
J LIBRARY OF CONGRESS.* 

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UNITED STATES OF AMERICA. 




Zo.lfro 









/ 



THE 



ELEMENTS OF ASTRONOMY 



OR 



THE WORLD AS IT IS, 



AND 






AS IT APPEARS. 



BY THE AUTHOR OF " THEORY OF TEACHING," " EDWARD'S 
FIRST LESSONS IN GRAMMAR," ETC. 



BOSTON: 

PUBLISHED BY CROCKER AND BREWSTER, 
47, Washington-street. 

1850. 



Entered according to Act of Congress, in the year 1850, 

BY CROCKER AND BREWSTER, 

In the Clerk's Office of the District Court of Massachusetts. 









PREFACE. 



The apology for the present treatise on Astronomy is 
based on the nature of the subject ; it is one which re- 
quires to be presented to the student from more than one 
point of view. He should learn from the original observer, 
the profound generalizer, the investigator of cause and 
effect in detail ; and to prepare him for such studies, he 
needs the book of one who knows from experience his re- 
quirements and his capacity. 

As such a book, " The Elements of Astronomy, or the 
World as it is, and as it appears," is offered by a teacher 
to the teaching and studying public. Had the writer 
aimed only to excite an interest in the subject, it would 
have been shorter and more attractive ; but it is intended, 
likewise, to exercise the student's memory, reason, and 
imagination. The details introduced for this purpose serve 
also to keep each truth before the mind some time ; they 
present it in different lights, and secure its being perceived 
by each pupil fully and in all its bearings. 



IV PREFACE. 

This book has gradually grown out of lessons given 
orally during many years of teaching. These were written 
out for the author's own use, not for publication. Origi- 
nality was not sought for, and all explanations and illustra- 
tions which could be of service to the pupils were adopted. 
As time passed on, and no book appeared precisely suited 
to the wants of these pupils, or of High Schools in general, 
the author began to entertain the idea that these lessons 
might in some measure supply the want so extensively felt. 
In this hope such completeness has been given to the work 
as a very limited leisure would allow. It has been revised 
in manuscript by George P. Bond, Esq., of the Cambridge 
Observatory, to whom the author is also indebted for 
superintending its passage through the press. 

Boston, 1850. 



CONTENTS 



Page. 
Chapter I. — Definitions. The Sphere. Spherical Distances and 
Angles. The Spheroid. Earth's Axis and Equator. Terrestrial 
Latitude and Longitude. The Sphere of the Heavens, Zenith and 
Horizon. Celestial Pole and Equator. The Ecliptic. The Zodiac. 
Right Ascension and Declination. Celestial Longitude and Lati- 
tude. Parallax. Terms defining the Orbit of a Planet. Sidereal 
and Apparent Time. 13 

Chapter II. — Light and the Telescope. Nature and properties of 
Light. Velocity of Light. Transmission and Reflection. Lenses, 
and the Refracting Telescope. Magnifying Power. Fields of View. 
Distinction between the different powers of a Telescope. Reflecting 
Telescope. Achromatic Telescope. 25 

Chapter III.— Astronomical Instruments. Difficulties in the con- 
struction of Telescopes. Telescope Stands. Transit Instrument. 
Graduated Circle and Vernier. Mural Circle. Polar and Horizontal 
Point. Transit Circle. The Equatorial. The Altitude and Azimuth 
Instrument. Theodolite. Sextant. Difficulties in Observing. Per- 
sonal Equation. Lord Rosse's Reflectors. 34 

Chapter IV.— Nebular and Sidereal Systems. The Milky way. 
Comparative dimensions of the Solar and Sidereal Systems. Dis- 
tances of the Fixed Stars. Classification of Stars according to their 
apparent Magnitudes. Distribution of the Stars. Gauging of the 
Heavens by Herschel. True form of the Milky Way. Clusters and 
Nebulse. Forms and distribution of Nebulae. Vastness of the Uni- 
verse. Effect of the finite velocity of Light. 48 

Chapter V.— Interior of our Sidereal System. Absolute and Relative 
Motion. Motions of the Fixed Stars. Proper Motion of 61 Cygni, 
and of Arcturus. Motion of the Solar System. Investigations of 
Herschel, Struve and Argelander. The Central Sun. Double and 
Multiple stars. New Stars. Variable Stars. Color of Stars. 61 



VI CONTENTS. 

Page- 
Chapter VI. — The Solar System. The Primary and Secondary- 
Planets. Law of distances from the Sun. Revolutions of the 
Planets in their Orbits, and their revolutions on their Axes. Ele- 
ments of their Orbits. Amount of Solar heat received by each 
Planet. Orbits described by the Secondary Planets. Mass and 
Densities of the Sun and Planets. Eclipses. General views of 
the Solar System. 74 

Chapter VII. — Meteors and the Zodiacal Light. Appearance and 
number of Meteors ; their composition and size. Meteoric show- 
ers ; their supposed origin. The Zodiacal Light ; its appearance ; 
different theories of its nature ; its possible connection with Mete- 
oric showers. 93 

Chapter VIII. — Comets. The number of recorded Comets. Variety 
in their motions and appearances. Their immense size. Descrip- 
tion of a Comet. The tails of Comets. Bessel's Theory of their 
formation. Halley's Comet. Biela's and Encke's. Their resistance 
by the ether. The mutual influence of Comets and Planets. 
Mass of the Comet of 1770. The probable effect of a collision with 
a Comet. 101 

Chapter IX. — Physical Astronomy. Analogies observable among 
the Planets. Their general form and their Atmospheres. Inter- 
nal state of our Globe. Central Heat. Theories accounting for 
the external appearance of the Earth and Moon. Objections to the 
theory of Central Heat. Supposed differences of temperature in 
space. Laplace's Nebular Theory. 117 

Chapter X.— The Rotation and Figure of the Earth. Effect of the 
Motion of the Earth on the apparent Motion of the Heavens. Dis- 
tinction between the Earth's yearly and diurnal Motion. Axis 
of Rotation. The Circles of the Equator and Ecliptic. Shape of 
the Earth's Orbit. Invariability of the Earth's Rotation. Earth's 
Figure and Dimensions. Its Ellipticity. Pendulum experiments. 
Measurement of Arcs of a Degree on the Earth's surface. 134 

Chapter XI — General Phenomena on the Earth's Surface. Univer- 
sal diffusion of Gravity over the Earth's Surface. Determination of 
the Earth's Mass and Density. Our Knowledge of the Earth's 
Surface. The Sea. Tides. Stability of the Ocean's Equilibrium. 
The Atmosphere. Clouds. Winds. Trade Winds. Use of the 
Atmosphere. Absorption and Diffusion of Light and Heat. Re- 
fraction. Twilight. 161 



CONTENTS. Vll 

Page. 

Chapter XII. — Phenomena which differ in different parts of the Earth. 
Day and Night. Circle of Illumination. Twilight The Seasons. 
Curve traced by the Sun's combined daily and yearly Motions. 
Portion of the Heavens visible in different Latitudes. Length of 
Day and Night. Equinoxes. Effect of Twilight. Amount of Light 
and Heat received in a given place. Equality of the distribution of 
Heat in the Northern and Southern Hemispheres. Difference in 
their respective Seasons. 187 

Chapter XIII. — Position of PI a,ces on the Earth and of Stars in the 
Heavens, Modes of defining position on the Earth's Surface. 
Methods of finding Latitude. Longitude. Its determination by 
the Moon's motion. The Sextant. Eclip-es of Jupiter's Satellites. 
Determination of Local Time. Lunar Distances. The Theodolite. 
Celestial Globes and Maps. Apparent Motions of the Planets. 
The Fixed Stars. The Zodiac. The Constellations. The Milky 
Way. Proposed Revision of the Constellations. 211 

Chapter XIV — Laws of Shape and Motion. Attraction of Gravita- 
tion. Effect of Gravitation on the figures of the Sun and Planets. 
The Figure of the Earth that of Equilibrium. Illustration of the 
effect of Rotation on a Fluid Mass. Laws of Gravity. Centre of 
Gravity. 238 

Chapter XV. — Laws of Motion, (continued}. Three general Laws 
of Motion. Composition of Forces. Path of a Projectile near the 
Earth's Surface. Motion in a Curve. Projectile and Centripetal 
Forces. Motion in the Solar System. Kepler's Laws. Central 
Forces. 249 

Chapter XVI. — Perturbations. Disturbing Forces. Problem of the 
three bodies. Stability of the Solar System. Periodical and Secu- 
lar Inequalities. Perturbations in Longitude. Motion of the Line 
of Apsides. Variation of the Eccentricities. Perturbations in Lati- 
tude. Retrogradation of the Nodes. Variation of the Inclinations. 
Permanency of the Major Axes. Effect of a Resisting Medium. 
Invariable Plane of the Solar System. Inequality in the Theory of 
Jupiter and Saturn. 267 

Chapter XVII — Precession, Nutation, and Aberration. Action of 
the Planets on the Plane of the Ecliptic. Action of the Sun and 
Moon on the Earth's Equator. The Precession of the Equinoxes. 
Motion of the Earth's Axis. Nutation. Aberration. Its Effect on 
the Apparent Places of the Stars. Methods of Computing it. Aber- 
ration of the Fixed Stars and of the Planets. 282 



Vlll CONTENTS. 

Page. 
Chapter XVIII — Time. Natural Divisions of Time. The Solar and 
Sidereal Day. Mean and Apparent Time. The Equation of Time. 
Variation in the Length of the Seasons. The Sidereal, Equinoctial 
and Anomalistic Years. Leap Year. Further Divisions of Time. 296 

Chapter XIX. — Parallax. Parallax defined. Horizontal Parallax. 
Methods of correcting for Parallax. Determination of the Moon's 
Parallax. Transits of Mercury and Venus. Methods of computing 
the Solar Parallax. Distances of the Sun and Planets. Parallax of 
the Fixed Stars. 307 

Chapter XX. — The Sun and Planets. The Mass and Dimensions 
of the Sun. Its Atmosphere. Its light and Heat. The Solar Spots. 
Proposed explanations of their Appearance. The Sun's Rotation. 
The Centre of Gravity of the Solar System. Determination of the 
Orbits, Masses and Densities of the Planets. 325 

Chapter XXI — The Planets, {continued). Mercury. Its rare visi- 
bility. Its Phases. Venus. Its brilliancy, Rotation and Atmo- 
sphere. Mars. Its Polar Spots. The Asteroids. Conjecture as to 
their Origin. Jupiter. Its Belts and Rotation. The Satellites of 
Jupiter. Theory of their Motion. Saturn. Its Rings and Satellites. 
Uranus. Neptune. 335 

Chapter XXII— The Moon. Size and Mass of the Moon. Its Dis- 
tance and Period. Revolution of the Nodes of the Lunar Orbit. 
Appearance of the Moon. Libration. Phases of the Moon. The 
Harvest Moon. The Lunar Theory. Action of the Sun. Evection. 
Variation. Annual Equation. Action of the Planets. Acceleration 
of the Mean Motion. 350 

Chapter XXIII. — Eclipses. Conditions necessary for an Eclipse. 
Lunar Eclipses. Dimensions of the Earth's Shadow. Limiis of a 
Lunar Eclipse, Solar Eclipse. Effect of the Moon's Parallax. 
Limits of a Solar Eclipse. Number of Eclipses in a Year. Eclipse 
of 1706. Eclipse of 1842. 367 



ELEMENTS OF ASTRONOMY. 



CHAPTEE I. 

DEFINITIONS. 

The Sphere. Spherical Distances and Angles. The Spheroid. Earths 
Axis and Equator. Terrestrial Latitude and Longitude. The Sphere of 
the Heavens, Zenith and Horizon. Celestial Pole and Equator. The 
Ecliptic. The Zodiac. Right Ascension and Declination. Celestial 
Longitude and Latitude. Parallax. Terms defining the Orbit of a 
Planet. Sidereal and Apparent Time. 

§ 1. A sphere is a solid terminated by a curved sur- 
face, all the points of which are equally distant from a 
point within called the centre. 

The radius of a sphere is a straight line drawn from the 
centre to a point in the surface ; the diameter, or axis, is 
a line passing through the centre, and terminated each 
way by the surface. 

All the radii of a sphere are equal ; all the diameters 
are also equal, and double the radius. 

§ 2. Every section of a sphere made by a plane is a 
circle. 

If a sphere is cut by a plane which passes through the 
centre, the section is called a great circle of the sphere ; 
the radius of such a section being the greatest possible, 
the same, namely, with the radius of the sphere. 

From this definition it is evident that a great circle may 
be made to pass through any two points in the surface of a 
sphere ; and that, if the two points be not opposite ex- 
tremities of a diameter, only one great circle can be made 
2 



14 ELEMENTS OF ASTRONOMY. 

to pass through them, for its plane must pass through the 
centre of the sphere, and only one plane can be made to 
pass through three points which are not in the same 
straight line. But through the two extremities of a diam- 
eter, any number of great circles may be made to pass, 
for they are in the same straight line with the centre of 
the sphere. 

§ 3. A diameter of a sphere, perpendicular to the plane 
of any great circle, is called the axis of that great circle ; 
and the extremities of the axis are called its poles. 

The angles formed at the centre of the sphere by the 
plane of a great circle and its axis, are right angles ; 
therefore the pole of a great circle is 90° distant from 
every point of the circumference of the great circle. The 
arcs subtending the angles are 90°, and are those of great 
circles ; and all angular distances on the surface of a 
sphere, to an eye at the centre, are measured by the arcs 
of great circles. 

All great circles bisect one another ; for all passing 
through the centre of the sphere, their common section 
must be a diameter of each, and every diameter bisects a 
circle. 

§ 4. Secondaries to a great circle are great circles 
which pass through its poles, and whose planes are there- 
fore perpendicular to its plane. Hence every secondary 
bisects its great cicle. If it passes through the poles of 
two great circles, it is perpendicular to each of them and 
bisects them both. And conversely, if one great circle be 
perpendicular to two others, it must pass through their 
poles. 

If an eye be in the plane of a circle, that circle appears 
a straight line ; hence in the representation of the surface 
of a sphere upon a plane, those circles whose planes pass 
through the eye are represented by straight lines. 

The angle formed by the circumferences of two great 
circles on the surface of a sphere is equal to the angle 
formed by the planes of those circles ; and is measured by 
the arc of a great circle, intercepted between them, and 
which is a secondary to each. 

§ 5. If a sphere is cut by a plane which does not pass 



ELEMENTS OF ASTRONOMY. 15 

through the centre, the section is called a small circle of 
the sphere ; the radius of such a section being less than 
that of the sphere. 

A circle, it is plain, may be made to pass through any 
three points in the surface of the sphere ; and it "will be a 
great or a small circle, according as its plane passes 
through the centre of the sphere or otherwise. 

Parallel circles of a sphere are such as have their planes 
parallel. Parallel circles have the same axis and poles ; 
for a straight line "which is perpendicular to one of two par- 
allel planes is perpendicular to the other likewise. Two 
parallel circles cannot both of them pass through the cen- 
tre of the sphere ; that is, they cannot both be great cir- 
cles of the sphere. 

If with the intersection of two great circles as a pole, 
a great circle be described, and also a small circle parallel 
to it, the arcs of the great and small circles intercepted 
between the two great circles contain the same number of 
degrees. And any one of these arcs measures the angle 
at the pole made by the planes of the two great circles. 

The centres of parallel circles lie in the diameter per- 
pendicular to their planes. 

§ 6. Either pole of a circle of the sphere is equally 
distant from ail points in the circumference of that circle ; 
whether the direct or spherical distance be understood. 

Hence any circle of a sphere may be conceived to be 
described from either of its poles as a centre with the 
spherical distance of that pole as a radius. For if this 
distance be carried round the pole, its extremity will lie in 
the circumference of the circle. 

The distances of any circle from its two poles are, to- 
gether, equal to a semi circumference. 

A great circle is equally distant from its two poles ; but 
this is not the case with a small circle. 

Equal circles of the sphere have equal polar distances, 
and conversely. 

The polar distances of any circle of the sphere are the 
spherical arcs which join any point in the circumference 
with the two poles of the circle. By the polar distance 
(singly) the lesser of these two arcs, or distance from the 
nearer pole, is generally to be understood. 



16 ELEMENTS OF ASTRONOMY. 

§ 7. Any portion of the circumference of a great cir- 
cle is called a spherical arc. 

Two points are said to be joined on the surface of the 
sphere when the spherical arc between them is described ; 
and this arc is called the spherical distance of the two 
points, in order to distinguish it from their direct distance, 
which is the straight line which joins them. The spherical 
distance of opposite extremities of a diameter of the sphere 
is evidently half the circumference of a great circle ; but 
the spherical distance of any other two points is less than 
a semi-circumference, being always the lesser of the two 
arcs into which they divide the great circle which passes 
through them. 

§ 8. If the arcs of two great circles meet in one point, 
they are said to form at that point a spherical angle. 

A spherical angle is greater or less according to the 
opening between its containing arcs. 

Every spherical angle is measured by the plane angle, 
which measures the inclination of the planes of the con- 
taining arcs. 

When one spherical arc, standing upon another, makes 
the adjacent spherical angles equal one to another, each of 
them is called a spherical right angle, and the arc which 
stands upon the other is said to be perpendicular, or at 
right angles to it. 

The terms acute and obtuse are likewise applied to 
spherical angles. 

Circles are thus said to make right, acute, or obtuse an- 
gles with one another. And we may measure this angle 
by the spherical angle on the surface made at the points 
where the circles intersect, or by the angles made by the 
planes of the intersecting circles, or by the angles made 
by the tangents of the two circles at the point of inter- 
section. It will be the same in whichever way we meas- 
ure it. 

If we draw from the pole of a great circle to any point 
in its circumference a spherical arc, this arc is a quadrant, 
or 90° of a great circle, and is at right angles to the cir- 
cumference. For since the pole is always 90° from the 
circumference, the axis and plane of the circle have an arc 



ELEMENTS OP ASTRONOMY. 17 

of 90° for the measure of their angle, therefore they make 
a right angle. 

If there be two equal and parallel small circles, and a 
great circle meets one of them in any point, it will meet 
the other in the opposite extremity of the diameter which 
passes through that point. 

If a great circle cuts one of two equal and parallel small 
circles, it will cut the other likewise. 

§ 9. In order to compare together different arcs and 
angles, every circumference of a circle is supposed to be 
divided into 360 equal arcs, called degrees, and marked 
thus, (°). For instance, 60° is read 60 degrees. 

Each degree is divided into 60 equal parts called min- 
utes, and marked ('). 

Each minute is divided into 60 equal parts called sec- 
onds, and marked ("). 

§ 10. As all circumferences, whether great or small, 
are divided into the same number of parts, it follows that 
a degree which is thus made the unit of arcs is not a fixed 
value, but varies for every different circle. It merely ex- 
presses the ratio of an arc, namely, -g-lu, to the whole cir- 
cumference of which it is a part, and not to any other. 

An angle has a fixed value altogether independent of 
the radius of the arc by which it is measured. But what- 
ever radius we give the arc, the arc will always have the 
same proportion to its circle, and this proportion gives the 
same number of degrees for the measure of the angle. 

If we make in the edge of a ruler five notches, and turn 
the ruler round one of its ends as a centre, making the 
ruler a radius, and describing a circle with its outer end, the 
five notches will have described five circles, and the ruler 
has made four right angles ; each of these angles having 
90° for its measure. 

§11. A tangent to a circle is a line which has only 
one point in common with the circle, and is perpendicular 
to a radius of the circle drawn to the point of contact. 

A tangent to a sphere has only one point in common 
with the sphere. 

A plane is tangent to a sphere when it touches its sur- 
face only at one point. Owing to the form of a sphere, a 
2* 



18 ELEMENTS OF ASTRONOMY. 

plane surface can touch its surface onlj at one point, \m" 
less it cuts the sphere. 

§ 12. If we suppose a sphere to be flattened at the 
poles, we shall have the solid called an oblate spheroid. 
Its shortest diameter will be through its poles ; its longest 
diameter will lie in the plane at right angles to its axis. 
All other diameters will be longer than the former of these, 
and shorter than the latter. Only one section of it will be 
a great circle, — the one 90° from its poles. There may 
be many small circles parallel to this. 

If we suppose the sphere to be lengthened out at the 
poles, it will be a prolate spheroid. 

We may suppose a sphere to be generated by the revo- 
lution of a circle about its diameter. 

An oblate spheroid is formed by the revolution of an 
ellipse about the shorter axis. A prolate spheroid is 
formed by the revolution of an ellipse about its longer 
axis. 

§ 13. The axis of the earth is that diameter about 
which it rotates with a uniform motion from west to east. 
The extremities of this diameter or the points where it 
meets the earth's surface are called the poles of the earth. 

The terrestrial equator is a great circle on the earth's 
surface equidistant from its poles, dividing it into two 
hemispheres, — a northern and a southern. The plane of 
the equator is therefore a plane perpendicular to the 
earth's axis, and passing through its centre. 

The terrestrial meridian of a place on the earth's sur- 
face is a great circle passing through both the poles and 
through the place. 

§ 14. The latitude of a place on the earth's surface is 
its angular distance from the equator. This angle lies at 
the centre of the earth, but is measured on the meridian 
of the place. It is reckoned in degrees, minutes, and sec- 
onds, northward or southward, according as the place lies. 

Parallels of latitude are small circles on the earth's sur- 
face parallel to the equator. Every point in such a circle 
has the same latitude. 

The longitude of a place on the earth's surface is the 
angle made by its meridian with the meridian of some 



ELEMENTS OE ASTRONOMY. 19 

place selected as a point to reckon from. Greenwich, in 
England, is usually taken for this point, reckoning 180° 
West and 180° East from Greenwich. 

Longitude is also reckoned in time. For as the earth 
rotates 360° in twenty-four hours, it follows that it rotates 
15° in one hour. In this case it is reckoned westward all 
round the globe. We need only divide the number of de- 
grees of longitude by fifteen and we have the number of 
hours. 

§ 15. The sphere of the heavens is an imaginary con- 
cave sphere of infinite radius, having the centre of the 
earth, or w T hat comes to the same thing, the eye of any 
spectator on the earth's surface, for its centre. Every 
point in this sphere may be regarded as the vanishing point 
of all the lines parallel to that radius of the sphere which 
passes through it, seen in perspective from the earth. 
Every great circle on it is the vanishing line of a system 
of planes parallel to the plane which passes through it and 
through the spectator's eye. 

§ 16. The zenith is the point of the sphere of the 
heavens vertically above the spectator ; the nadir, the point 
180° distant under his feet. They are therefore the vanish- 
ing points of all lines parallel to the direction of a plumb 
line at his station. The plumb line itself is, at every point 
of the earth, perpendicular to its spherical surface. At 
no two stations, therefore, can the actual directions of two 
plumb lines be regarded as mathematically parallel ; they 
converge towards the centre of the earth. But for very 
small intervals, (as in the area of a building, or in the 
same town,) the difference from exact parallelism is so 
small that it may be practically disregarded. An interval 
of a mile gives to plumb lines a convergence of about a 
minute. 

§ 17. The zenith and nadir are the poles of the celes- 
tial horizon ; that is, they are points 90° distant from every 
point in it. The celestial horizon is the vanishing line of a 
system of planes parallel to the sensible and the rational 
horizon. 

The sensible horizon is the actual horizon of the spectator. 
If we suppose a plane to be extended from a point on the 



20 ELEMENTS OF ASTRONOMY. 

earth's surface to the sphere of the heavens, and another 
plane parallel to the former to be extended from the 
earth's centre to the sphere of the heavens, these two 
planes will cut the sphere in the same line. The first of 
these planes is called the sensible, the latter the rational 
horizon, and both coincide with the celestial horizon. 

Vertical circles of the sphere are great circles passing 
through the zenith and nadir, or great circles perpendicu- 
lar to the horizon. On these are measured the altitudes 
of objects above the horizon, — the complements to which 
are their zenith distances. 

§ 18. The poles of the heavens are the points of the 
sphere to which the earth's axis is directed ; or the van- 
ishing points of all lines parallel to the earth's axis. The 
star nearest to each celestial pole is called the pole star. 

The celestial equator, or equinoxial, is a great circle of 
the heavens marked out by the indefinite extension of the 
plane of the terrestrial equator, and is the vanishing line 
of all planes parallel to it. 

§ 19. When the plane of the terrestrial meridian of a 
spectator is prolonged to the sphere of the heavens, it 
marks out the celestial meridian of a spectator stationed at 
that place. The intersection of the spectator's meridian 
with his horizon makes its north and south points. 

The vertical circle which cuts the meridian of any place 
at right angles, is called the prime vertical ; and the points 
where it cuts the horizon are called the east and west 
points. Hence the east and west points are 90° distant 
from the north and south points. These four are called 
the cardinal points. 

§ 20. Azimuth is the angular distance of a celestial 
object from the north or south point of the horizon, re- 
ferred to the horizon by a vertical circle ; or it is the an- 
gle comprised between two vertical planes ; one passing 
through the elevated pole, the other through the object. 
The altitude and azimuth of an object being known its 
place in the visible heavens is determined. 

§ 21. The ecliptic is that great circle in the heavens 
which the earth really describes, and which the sun ap- 
pears to describe in the course of the year. 



ELEMENTS OF ASTROXOMY. 21 

The ecliptic and the equator, being great circles, must 
bisect each other, and their inclination is called the ob- 
liquity of the ecliptic. The points "where they intersect are 
called the equinoxial points. The times when the sun 
comes to these points, and the points themselves, are called 
the equinoxes, because the day and night are then equal 
all over the world. 

§ 22. The ecliptic is divided into twelve equal parts 
called signs : — Aries, Taurus, Gemini, Cancer, Leo, Yirgo, 
Libra, Scorpio, Sagittarius, Capricornus, Aquarius, Pisces. 
The order of these is according to the real motion of the 
earth, and the apparent motion of the sun. The first point 
of Aries coincides with the position of the sun at the vernal 
equinox, the first point of Libra with the position of the 
sun at the autumnal equinox. 

The first six signs are called northern, lying on the 
north side of the equator ; and the last six are called 
southern, lying on the south side. 

When the motion of the heavenly bodies is according 
to the order of signs it is called direct. When it is in a 
contrary direction it is called retrograde. The real mo- 
tion of all the planets is according to the order of signs, 
but their apparent motion is sometimes in an opposite di- 
rection. 

The zodiac is the zone extending on each side of the 
ecliptic, within which the motion of the planets, and the 
moon, and the apparent motion of the sun, are performed. 

§ 23. When a body is referred to the equinoxial, its 
spherical distance from that is called its declination. 

The arc in the heavens which corresponds to terrestrial 
longitude is called an arc of right ascension. As terres- 
trial longitudes are reckoned from a determinate point on 
the equator, so right ascensions require some known point 
in the equinoxial as the commencement of their reckoning, 
or their zero point. Some hour-circle or meridian, of the 
celestial sphere, must be taken for their starting point. 
The most obvious, indeed, the only point naturally marked 
on the equinoxial, is the one where the ecliptic intersects it, 
and though this is not exactly stationary, it has been 
adopted as the zero point. It serves also as a zero point 



22 ELEMENTS OF ASTRONOMY. 

for reckoning on the ecliptic. The right ascensions are al- 
ways reckoned eastward from the equinox in degrees or in 
hours. 

The longitude of a star is an arc of the ecliptic inter- 
cepted between the first point of Aries, and a secondary 
to the ecliptic passing through the star. If the body be in 
our system, and seen from the sun, it is called the helio- 
centric-longitude ; if seen from the earth, it is called the 
geocentric longitude ; the body being in each case referred 
perpendicularly to the ecliptic by a plane passing through 
the eye. 

The latitude of a star is its angular distance from the 
ecliptic, measured upon a secondary to the ecliptic drawn 
through the star. If the body be in our system, its angu- 
lar distance from the ecliptic seen from the earth, is called 
the geocentric latitude ; but if seen from the sun, it is 
called the heliocentric latitude. 

§ 24. The tropics are two parallels of declination, 
touching the ecliptic. One, touching it at the beginning 
of Cancer, is called the tropic of Cancer ; and the other, 
touching it at the beginning of Capricorn, is called the 
tropic of Capricorn. The tropics are so called from a 
Greek word signifying to turn, because the sun appears to 
turn there and descend or ascend in the heavens. The 
two points where the tropics touch the ecliptic are called 
the solstitial points, because the sun appears to linger or 
stand still there. The times when the sun is seen in these 
points are called the solstices. 

The colures are two secondaries to the celestial equator ; 
one, passing through the equinoctial points, is called the 
equinoctial colure ; the other, passing through the solstitial 
points, is called the solstitial colure. 

The arctic and antarctic circles, (so named from Arctos, 
the bear, and anti-Arctos, opposite the bear,) are two paral- 
lels of declination ; the former about the north, and the 
latter about the south pole, the distances of which from 
the two poles are equal to the distances of the tropics from 
the equator. These are also called the polar circles. 

§ 25. The altitude of the elevated pole being equal 
to the spectator's geographical latitude, it follows that all 



ELEMENTS OF ASTRONOMY. 23 

the stars whose polar distance does not exceed his latitude 
are perpetually visible. A circle described round the ele- 
vated pole, with a radius equal to the altitude, is called a 
circle of perpetual apparition. A circle of equal size 
round the depressed pole is called a circle of perpetual 
occupation, because the stars within it are never visible. 

All celestial objects within the circle of perpetual appa- 
rition come twice on the meridian above the horizon in 
every diurnal revolution ; once above and once below the 
pole. These are called their upper and lower culmina- 
tions. 

All the heavenly bodies culminate, (i. e. come to their 
greatest altitudes,) on the meridian, which is, therefore, 
the best situation to observe them, as they are there least 
confused by the inequalities and vapors of the atmosphere, 
as well as least displaced by refraction. 

§ 26. A body is in conjunction with the sun when it 
has the same longitude ; in opposition when the difference 
in their longitudes is 180°, and in quadrature when the 
difference in their longitudes is 90°. 

Syzygy is either conjunction or opposition. 

The elongation of a body from the sun is its angular dis- 
tance from the sun when seen from the earth. 

The diurnal parallax is the difference between the appa- 
rent places of bodies as seen from the centre and from the 
surface of the earth. 

The annual parallax is the difference between the appa- 
rent place of a body when seen from opposite points of the 
earth's orbit. 

§ 27. The points where the orbits of the planets cut 
the ecliptic, and where the orbits of the secondaries cut 
the orbits of their primaries, are called their nodes. That 
node is called ascending where the planet passes from the 
south to the north side of the ecliptic ; the other is called 
the descending node. The line which joins the nodes is 
called the line of the nodes. 

Aphelion is that point in the orbit of a planet which is 
farthest from the sun. 

Perihelion is that point in the orbit of a planet which 
is nearest to the sun. 



24 ELEMENTS OF ASTRONOMY. 

Apogee is that point of the moon's orbit which is farthest 
from the earth. 

Perigee is that point of the moon's orbit which is nearest 
to the earth. 

The apsis of an orbit is either its aphelion or perihelion, 
apogee or perigee ; and the line which joins the apsides is 
called the line of the apsides. 

The true anomaly of a planet is its angular distance at 
any time from its aphelion or apogee. The mean anomaly 
is the angular distance from the same point at the same 
time, if it had moved uniformly with its mean angular ve- 
locity. 

The equation of the centre is the difference between the 
true and the mean anomaly. 

The mean place of a body is the place where a body 
(not moving with a uniform angular velocity about the 
central body) would have been, if it had moved with its 
mean angular velocity. The true place of a body is the 
place where the body actually is at the time. 

The corrections applied to the mean place of a body in 
order to get its true place are called equations. 

§ 28. Apparent noon is the time when the sun comes 
to the meridian. True or mean noon is twelve o'clock by 
a clock adjusted to go twenty-four hours in a mean solar 
day. The equation of time at noon is the interval between 
true and apparent noon. 

Sidereal time is reckoned by the apparent diurnal mo- 
tion of the stars, or rather of that point in the equinoctial 
from which right ascensions are reckoned. The interval 
between any two successive returns of this to the meridian 
is called a sidereal day, and is divided into twenty-four si- 
dereal hours. 



ELEMENTS OF ASTRONOMY. 25 

CHAPTER II. 

LIGHT AND THE TELESCOPE. 

Nature and properties of Light. Velocity of Light. Transmission and; 
Reflection. Lenses, and the Refracting Telescope. Magnifying Power. 
Fields of View. Distinction between the different powers of a Tele- 
scope. Reflecting Telescope. Achromatic Telescope. 

§ 29. Philosophers are not agreed about the nature 
of light. Some maintain that it is an emanation from 
luminous bodies ; others suppose it to be produced like 
sound, bj the undulations of a subtile fluid diffused 
throughout space. Many facts favor this last theory. 
Bays of light cross in every direction without interfering ;. 
and we can suppose this more easily of waves than of 
actual particles. If light is a substance, what becomes of 
it ? where is all that has been emitted since the beginning 
of the world ? If it is merely a motion of particles, it 
may cease like any other motion. As the undulatory the- 
ory has much in its favor, we shall adopt it and consider 
light as that vibration of an unknown fluid which causes to 
our eye the sensation of sight. We cannot attain perfect 
certainty as to the nature of light, but we can learn its 
laws, and reason concerning them without this certainty. 

§ 30. Bodies are divided into luminous and non-lumin- 
ous. All light comes originally from some self-luminous 
body. It is emitted from every point of such a body, and 
in every direction in which the point is visible. A ray is 
a single line of light ; a pencil of rays is a collection pro- 
ceeding from any one point of such a body. These rays 
move through a uniform medium in straight lines, entirely 
independent one of another. We know that rays move in 
straight lines, for no opaque body can screen a luminous 
point from us unless interposed in a straight line between 
us and it. Light moves with a velocity of 192,500 miles 
in a second. It travels from the sun to the earth in eight 
minutes ; a distance which the earth, moving nineteen 
miles a second, will take two months to pass through. 
3 



26 ELEMENTS OF ASTRONOMY. 

It moves through a space equal to the circumference of 
our globe in the eighth part of a second, a flight which the 
swiftest bird could not perform in less than three weeks. 
Light diminishes in intensity according to the squares of 
the distance. At a certain distance let it be diffused over 
a certain space and have a certain intensity, at twice the 
distance it is diffused over four times the space, and has 
but one fourth the intensity. 

§ 31. Luminous bodies, probably from being them- 
selves in agitation, cause vibrations in some medium which 
excites our optic nerve, and makes us conscious of the ob- 
ject at which the vibration began. Hearing, which seems 
to be a coarser sense, is excited in the same manner. But 
we know that the air, wood, ice, rock, or almost any 
substance, may serve as a medium for the vibrations which 
affect the auditory nerve. While we cannot detect the 
medium which transmits vibrations to the optic nerve, as 
the transmission is so much more rapid than in case of 
sounds, and as we see to far greater distances than we 
hear, we cannot but feel it probable that the medium is 
more thin and subtile, and that no other can be substituted 
for it. This medium must extend to the stars, otherwise 
no vibrations from them could reach us ; it must exist in 
interstices of transparent bodies, such as air, water, and 
glass. Other facts also make it probable that regions be- 
yond our earth are occupied by a thin ether. The comets, 
whose frequent returns enable us to calculate their periodic 
times, seem to be drawn nearer and nearer to the sun, 
their times are diminishing. If so, they must meet some 
resistance which gradually diminishes their projectile 
power, and this resistance must come from an invisible 
thin fluid. A fluid too thin to act sensibly upon the plan- 
ets, but dense enough to derange the lighter comets, and 
to transmit light. 

§ 32. Non-luminous bodies are either opaque or trans- 
parent. When light falls on an opaque body, a part of it 
is reflected, a part enters the body and is lost. When the 
body is transparent, the greater part is transmitted, and 
but little is reflected. Both reflection and transmission fol- 
low particular laws, which must be understood previous to 
studying astronomy. 



ELEMENTS OP ASTRONOMY. 27 

Bodies are referred to the direction from which the ray 
by which we see them comes to our eyes. We cannot 
know, except by reasoning, what changes this direction 
may have undergone. 

§ 33. Reflected light serves as well as original light to 
show us the direction of bodies. We see the moon, and the 
clouds, we see objects around us by light which may have 
been many times reflected. Reflected light is however 
weak, because a great portion of that which falls on 
bodies is usually absorbed. There is one condition neces- 
sary to our seeing a body by reflected light. The angle 
by which the light passes to us is always equal to that by 
which it strikes upon the reflecting body. The angle of 
reflection is equal to the angle of incidence. We must, 
therefore, be in such a position that the ray reflected at 
this angle shall enter our eye, or we shall not see the 
body. 

§ 34. When light passes obliquely from a medium to 
one more dense, it is bent at the surface of the denser me- 
dium into a line more nearly approaching a perpendicular. 
It passes without bending through the medium, and if it 
again passes into the thinner medium, it is turned away 
from the perpendicular by an angle equal to that at which 
it entered. The more obliquely it enters the more it is 
turned aside ; the sine of the angle of incidence always bears 
the same ratio to the sine of the angle of refraction, when 
the same media are compared. When light passes per- 
pendicularly from one medium to another, it suffers no re- 
fraction. 

By interposing between the eye and the object a mate- 
rial of higher refractive power than air, and making it of 
such a form that the refraction at one or both surfaces 
shall bring the rays to a focus behind it, we bring the im- 
age of the object nearer to the eye, and thus increase its 
apparent size, brightness, and rapidity of motion. 

§ 35. The material used for the lenses of telescopes is 
glass. The lenses used are chiefly the plano-convex, the 
double convex and the double concave. Lenses of the same 
material converge the rays in proportion to the convexity of 
their form. A small portion of a large sphere refracts to 
a longer focus than a large portion of a small sphere. 



28 ELEMENTS OF ASTRONOMY. 

The simplest telescope consists of a plano-convex lens, 
whose focal length exceeds six inches, placed at one end 
of a tube whose length must be six inches greater than 
the focal length of the lens. Six inches being the dis- 
tance at which an eye sees small objects with ease. An 
eye placed at the other end of the tube will see an inverted 
image of distant objects magnified in proportion to the 
focal length of the lens. If the lens has a focal length of 
from ten to twelve feet, the magnifying power will be from 
twenty to twenty- four ; to a very shortsighted person, who 
sees objects distinctly at three inches, the magnifying 
power would be from forty to forty-eight. 

§ 36. It is well known that the same object brought 
nearer to the eye subtends a larger angle, and therefore 
appears larger than when more distant. It is also true 
that a small body, very near, subtends a larger angle, and 
therefore appears larger than a distant object of greater 
size. 

Let us examine the effects of placing convex lenses at 
different distances between the eye and an object. We 
will suppose a man to be 100 feet from the eye. We will 
place a convex glass of twenty-five feet focal distance half 
way between the man and the eye. The inverted image 
of the man will be formed fifty feet behind the lens, of the 
same size as the object. This image can be seen by the 
eye placed six inches behind it with great distinctness, and 
nearly as well as if the man had been brought from the 
distance of 100 feet to six inches. Thus the man, though 
seen no larger than life, would be apparently magnified, 
because his apparent magnitude would be increased in the 
proportion of 100 feet to six inches. If a lens of shorter 
focus be placed between them in such a way that the man 
is twenty feet before the lens, and his image eighty feet 
behind it, the size of the image would be four times that of 
the object. The man would be magnified four times di- 
rectly by the lens, and 200 times by being brought nearer 
to the eye ; in the whole 800 times. 

§ 37. When the focal length of the lens is quite incon- 
siderable compared with the distance of the object, as it is 
in viewing the heavenly bodies, the rule becomes this :— - 



ELEMENTS OF ASTRONOMY. 29 

Divide the focal length of the lens by the distance at which 
the eye looks at the image. If we use a lens with a focal 
distance of ten feet, on account of the sun's being 
95,000,000 miles distant, the image of the sun formed in 
the focus will be only ~^ ^miiea °^ ^e true s ^ ze °f tne 
sun's disc. But this image is viewed at a distance of six 
inches instead of 95,000,000, therefore it is magnified 
^ZToT^T (=20 times. 

95,000,000 miles X 6 inches. J 

§ 38. A lens placed in a telescope as we have des- 
cribed it, is called an object glass. It bestows on a tele- 
scope the power of bringing the image near, which is evi- 
dently the same thing as transporting the observer to the 
distant regions of space. But there is another way of in- 
creasing the apparent magnitude of objects, particularly of 
those which are within our reach, which is of great im- 
portance in optics. Rays from each point of a distant ob- 
ject enter the eye in parallel lines, and the object is dis- 
tinctly seen. If we bring an object very near, so as to 
give it great apparent magnitude, it becomes indistinct ; 
because the rays diverge from each point so much that the 
eye cannot bring them to a focus. But if we can make 
the rays from it enter the eye nearly parallel, it will be 
distinctly seen. Since parallel rays are brought to a focus 
by a lens, if we place an object or its image in the focus 
of the lens, the rays will come out from it parallel. We 
shall see the object very distinctly, and it will be magnified 
in the proportion of six inches, at which we see small ob- 
jects most distinctly, to its present short distance from the 
eye. But this short distance is equal to the focal length 
of the lens, so that the magnifying power produced by the 
lens is equal to six inches divided by the focal length of 
the lens. 

Such a lens is called the eyepiece of the telescope. 
These two lenses together constitute a simple form of the 
astronomical refracting telescope. 

§ 39. Let us consider the effect of each of these 

glasses upon the apparent motion and brightness of objects 

seen through them. Distant motions appear slow, because 

the space passed through subtends so small an angle at the 

3* 



30 ELEMENTS OF ASTRONOMY. 

eye. The moon wheels round the earth 2,000 miles an 
hour ; yet owing to her distance from it her motion is not 
visible to the naked eye. The same space described by two 
objects, one a hundred times more distant than the other, 
would seem a hundred times smaller in the former case 
than in the latter. The object-glass, since it brings objects 
nearer, must therefore increase their apparent motion in 
the same ratio. 

§ 40. A very rapid motion may be imperceptible, if 
the distance of the moving body be sufficiently great. 
For, the greater the distance, the smaller the angle under 
which the motion appears to the eye ; and if the arc which 
a body describes in an hour does not subtend an angle of 
more than 20°, its motion is imperceptible. The greatest 
apparent motion in the heavens does not exceed 15° in an 
hour, being that produced by the rotation of the earth. 
The greatest central motion is that of the moon and does 
not exceed about 13° a day. 

The apparent motion of heavenly bodies caused by the 
rotation of the earth is magnified by the object glass. 
The field of view appearing larger than it really is, bodies 
appear to pass through it in the same time more swiftly 
than they would to the naked eye. 

The eye piece also increases apparent motion in propor- 
tion to its power of magnifying. As in the solar micro- 
scope where the motions of animalcules appear prodigiously 
swift. 

Since the object and the eye glasses affect motion simi- 
larly, we shall in the following calculations consider not the 
power of each separately, but the whole power of the tele- 
scope. 

§ 41. Let a person direct the tubes of a telescope 
(without the lenses) to any celestial object, and there fix 
them ; he will soon find that in a short space of time the 
object will have removed from before the mouth of the 
tube. Now this motion of the celestial bodies, which is 
only apparent, arises from the rotation of the earth upon 
her axis ; and the quantity of this motion may be deter- 
mined with facility thus : the earth is known to revolve 
once about her axis in twenty-four hours, and as every circle 



ELEMENTS OF ASTRONOMY. 31 

is supposed to be divided into 360 equal parts or degrees, 
the apparent time any celestial body takes to describe one 
degree will be found by dividing the 24 hours by 360, 
which gives us four minutes as the time an object in the 
celestial equator would take to pass the mouth of the tube 
if it only takes in one degree of the heavens. 

In ordinary telescopes the field of view is about half a 
degree, and an equatorial star will cross it in two minutes. 
The moon's diameter is 30', it will therefore fill the field 
of such a telescope, and the whole moon, from the time 
one edge enters till the other leaves, will require four min- 
utes. If now we suppose the glasses to be put in the 
tube, the magnifying power will cause all bodies nearer 
than the fixed stars to look larger, and to appear to move 
faster, but the size of the tube remaining the same, no 
larger portion of the heavens from side to side can be 
seen through it than before. On the contrary, as a gen- 
eral rule, the higher the magnifying power the smaller are 
the linear dimensions of the field of view, while the appa- 
rent diameter and motion of objects is increased. 

§ 42. For convenience of calculation the linear dimen- 
sions have been used. But in truth when a surface is 
magnified it is so in all directions, and the increase of sur- 
face will be according to the square of the linear increase. 
Thus when the moon is brought by the telescope ten times 
nearer, its disc is one hundred times increased, and the 
area of the field of view one hundred times diminished. 
In the case of the fixed stars, while the telescope increases 
their brilliancy, no magnifying power that has yet been ap- 
plied has shown their true discs, owing to their immense 
distance from us. 

§ 43. There are three modes in use among astrono- 
mers of designating the capacity of a telescope. The first 
is by its illuminating power, which is proportioned to the 
area of its object glass ; the second by its magnifying 
power, which may be increased to any extent by changing 
the lenses of the eye pieces ; and the third is by its space- 
penetrating power, by which is meant such a relation be- 
tween the illuminating and the magnifying powers that the 
object viewed shall appear with its natural brightness. 



82 ELEMENTS OF ASTRONOMY. 

That is, of the brightness with which the object would ap- 
pear to the naked eye were the observer transported to a 
point as many times nearer to the object as the magnifying 
power employed is greater than unity. 

In all cases the space-penetrating power of a telescope 
is limited by the area of its object glass, or of its speculum, 
if it be a reflector. 

§ 44. On account of the difficulty of obtaining large 
lenses of a sufficiently pure material, a concave mirror is 
sometimes used instead of an object glass. The telescope 
is then called a reflector, as the rays of light are brought 
to a focus by reflection from the polished surface of the 
speculum. Large reflecting telescopes are not so well 
adapted to delicate observations and measurements as to 
discoveries in physical astronomy, the resolution of nebulae, 
and the examination of extremely faint objects. Those of 
Herschel and Lord Rosse are the most celebrated. In 
Herschel's forty foot reflector the observer turns his back 
to the object and looks in at the mouth of the telescope 
tube, near to the edge of which the image is thrown by a 
slight inclination of the mirror. Lord Rosse's telescope has 
a speculum of six feet diameter and a tube of sixty feet. 

§ 45. There is a difficulty both in convex lenses and 
concave mirrors arising from their shape, called spherical 
aberration. The rays from the central part of the lens or 
mirror are brought to a focus later than those from the 
edges and the rest of the surface. Spherical aberration 
causes indistinctness of vision by spreading out every mathe- 
matical point of the object into a small spot in its picture ; 
these spots by mixing with each other confuse the whole. 

It is avoided in mirrors by making the surface not 
spherical, but of the shape formed by the revolution of a 
parabolic curre. The rays reflected from the edges of the 
mirror are, on account of the form of the surface, brought 
to a focus as soon as those from the central portion. It is 
destroyed in refracting telescopes by combining a meniscus 
with a double convex lens, or by giving certain proportions 
to the figure of a single lens. 

§ 46. Rays from each point of the object spread over 
the whole lens and should be refracted to a point ; but if 



ELEMENTS OP ASTRONOMY. 33 

refraction is unequal in different parts of the lens, rajs 
from a point cannot be converged to a point, but either the 
more or the less converging will be spread out. 

There is also a difficulty in refracting telescopes, owing 
to the differing refractions of the different colored rays. 
Violet rays, being the most refrangible, come to a focus 
sooner than those of any other color. Red rays, being 
less so, will have a longer focus. Any image formed in 
the focus of violet rays will be violet colored, and images 
may be formed between this and the focus of red rays of 
each color of the spectrum. Therefore no white image 
can be formed, for there is no one spot in which all the 
colored rays will be present. 

§ 47. If the dispersive powers of different media were 
in proportion to their refractive powers, it would be impos- 
sible to correct this chromatic aberration. But fortunately 
different media produce spectra of different lengths when 
the mean refraction is the same. Let us compare two 
prisms, one of crown glass, the other of flint glass, with 
such a refracting angle that the light shall enter and quit 
them at equal angles, the mean ray of each will have the 
same refraction. But the spectrum produced by the flint 
glass will be longer than that produced by the crown 
prism. Thus flint glass is said to have a greater disper- 
sive power than crown glass, because at the same angle of 
mean refraction it separates farther the extreme rays of 
the spectrum. Diamond has a refraction nearly three 
times that of glass, while its dispersive power is less than 
that of glass. 

§ 48, ~Now if different media produce different bands 
of color with the same focal length, it follows that they 
may produce equal bands of color with different focal 
lengths. A concave lens may be used with a convex lens 
of equal dispersive power, but a higher refractive power, 
and the excess of the refractive power, will be the avail- 
able power of the lens, and white light will be refracted to 
the focus. Such a lens is called an achromatic lens, and. 
the image formed by it would be perfect were there not in 
equal spectra formed by different media a difference which 
prevents their entirely neutralising one another's refrac- 



34 ELEMENTS OF ASTRONOMY. 

tion. The bands of the same color in the two spectra are 
not of equal breadth, and therefore the images seen 
through such a lens are bordered on one side with a pur- 
ple, on the other with a green fringe. 

A telescope which is free from dispersion is called achro- 
matic ; one which is free from aberration also is called 
aplanatic, or free from all errors. 

Reflecting telescopes are perfectly free from color. 
For compound light is reflected, though not refracted, en- 
tire, all the colors following the same law of equal angles 
of incidence and reflection. 



CHAPTER III. 

ASTRONOMICAL INSTRUMENTS. 

Difficulties in the construction of Telescopes. Telescope Stands. Transit 
Instrument. Graduated Circle and Vernier. Mural Circle. Polar and 
Horizontal Point. Transit Circle. The Equatorial. The Altitude and 
Azimuth Instrument. Theodolite. Sextant. Difficulties in Observing. 
Personal Equation. Lord Rosse's Reflectors. 

§ 49. Though the theory of the construction of tele- 
scopes is attended with many difficulties, those which oc- 
cur in practice are as numerous, perhaps some of them are 
insuperable. For a reflector, a perfectly uniform metal is 
required, free from all microscopic pores, not liable to tar- 
nish, not so hard as to be incapable of taking a good figure 
and an exquisite polish, not so soft as to be easily scratched. 
Various compositions of metals are employed, consisting 
chiefly of copper and tin, with a little zinc, arsenic or sil- 
ver. After casting they must be ground and polished 
with the utmost care. Lenses also require the greatest 
nicety in composition, casting, grinding, polishing, and 
centering. To ascertain if the shape is perfect, an opaque 
back is placed behind the lens, the lens is made to revolve, 
and a lighted candle is brought before it, whose reflected 



ELEMENTS OE ASTRONOMY. 85 

image is attentively watched. If this image has any mo- 
tion, the lens is not perfect in its adjustment. 

§ 50. After an instrument is completed the next desi- 
deratum is a steady and immovable stand, free from vibra- 
tion. The instrument should be supported at both ends to 
give steadiness, and to prevent its being aifected by the 
wind ; for every vibration will be increased in the same 
ratio as the amplification of the instrument, and produce a 
tremulous or dancing motion in the objects. Thus a supe- 
rior telescope badly supported may be inferior to a com- 
mon one on an immovable stand. 

The materials of which stands are composed should be 
capable of transmitting as little vibration as possible ; the 
vibration of a frame of cast iron in one piece, though other- 
wise perfectly steady, would be sufficient to destroy dis- 
tinct vision. The difficulty of preventing vibrations in 
reflecting telescopes greatly impairs their value, as they 
are more affected by such disturbance than refractors. A 
telescope, which taken from its stand and placed on a lump 
of soft clay would enable a person to read a bill placed at 
a distance of 900 feet, would on its stand make it distinct 
only at a distance of 650 feet, although no tremor would 
be discerned on the stand. 

§51. A difficulty in placing an instrument arises from 
the absence of natural indications, other than those afforded 
by astronomical observations themselves, whether an in- 
strument has or has not its true position with respect to 
the horizon and its cardinal points, the axis of the earth, or 
to other principal astronomical lines and circles. For in- 
stance, to place a transit instrument correctly, we must 
know the direction of our meridian, but we must first learn 
this meridian approximately by observing the shadow cast 
by the sun at noon. 

The transit instrument consists of an astronomical tele- 
scope with wires and a micrometer, and is mounted on a 
nicely formed axis at right angles to itself. This axis re- 
maining always horizontal and directed to the east and 
west points of the horizon rests at its extremities in two 
sockets perfectly even, and set in two blocks of stone of a 
size and weight sufficient to prevent all agitation. The 



36 ELEMENTS OF ASTRONOMY. 

smooth extremities of the axis are capable of nice adjust- 
ment by screws, both in a vertical and horizontal direction. 
By placing a spirit level on the points, the axis can be 
made perfectly horizontal. Whether the axis lies precisely 
east and west can only be nicely ascertained by observa- 
tions made with the instrument itself. When it is per- 
fectly well adjusted the central line of the telescope will 
not quit the plane of the meridian when the instrument is 
turned round on its axis. The transit instrument is used 
to note the passage of bodies over the meridian, to note 
the right ascension of the fixed stars, the upper and lower 
culminations of the circumpolar stars, and for various prob- 
lems in time and longitude. 

§ 52. In the focus of the eye-piece and at right an- 
gles to the length of the telescope is placed the system of 
wires. This consists of one horizontal and five equidistant 
vertical threads or wires, which always appear in the field 
of view when properly illuminated, by day by the light of 
the sky, by night by that of a lamp. The horizontal wire 
is fixed, the middle vertical wire is brought to bisect the 
axis of the telescope, and thus to coincide with and repre- 
sent that portion of the celestial meridian which appears 
in the field of view. When a star crosses this wire it cul- 
minates, or passes the celestial meridian. The instant of 
this event is noted by a clock or chronometer, an indis- 
pensable accompaniment of the transit instrument. For 
greater precision, the moment of crossing each of the five 
or seven vertical threads is noted and a mean taken be- 
tween the times thus obtained, the threads being equidis- 
tant ; this tends to subdivide and destroy the errors. 

An important observation of its correctness consists in re- 
versing the ends of the axis, or turning it east for west. If 
this be done, and it gives the same results, we may be sure 
that the line of collimation of the telescope is at right an- 
gles to its axis, and marks out in the heavens a great circle. 

§ 53. To measure any small angular distance with 
a micrometer, as the diameter of a planet, two parallel 
wires are made to approach to or recede from each other 
till the body to be measured is exactly inclosed by them. 
Having accurately measured the planet by the two cross 



ELEMENTS OE ASTRONOMY. 3? 



The wires are moved by screws. The very slow motion 
which may be imparted to the end of a screw by a very 
considerable motion in the power, makes it very useful 
•in the measurement of minute motions and spaces. Sup- 
pose a screw cut so as to have fifty threads in an inch> 
each revolution of the screw will advance its point through 
the fiftieth part of an inch. Now suppose the head of the 
screw to be a circle whose diameter is one inch, the cir- 
cumference of the head will be 3.14 inches. This may 
easily be divided into a hundred equal parts distinctly visi- 
ble. If a fixed index be presented to this graduated cir- 
cumference, the hundredth part of a revolution of the 
screw may be observed by noting the passage of one di- 
vision of the head under the index. Since one entire revo- 
lution of the head moves the point through the fiftieth of 
an inch, one division will correspond to the five thousandth 
part of an inch. 

Micrometer threads are made of spiders' webs, India- 
rubber and glass threads, hair and wires. By thickly 
coating a fine platina wire with silver and drawing it out 
as fine as possible, and then dissolving the silver but not 
the platina, a very fine wire is obtained. 

§ 54. The angular intervals measured by means of 
the clock and transit instrument, are arcs of the equinoc- 
tial intercepted between the hour circles passing through 
the objects observed. Their measurement is performed 
by no artificial graduation of circles, but by the help of the 
earth's diurnal motion, which carries equal arcs of the 
equinoctial across the meridian, in equal times, at the rate 
of 15° per sidereal hour. 

In all other cases, w T hen angular intervals are to be 
measured, circles or portions of circles are referred to, 
others constructed of metal, and mechanically subdivided 
into equal parts, such as degrees, minutes, &c. The in- 
strument is sometimes movable upon the circle, sometimes 
both revolve together on an axis concentric with the circle, 
and forming one piece with it. As the telescope and cir- 
cle revolve through any angle, the part of the limb of the 
latter, which by such revolution is carried past the index, 
4 



38 ELEMENTS OP ASTRONOMY. 

will measure the angle described. The index may be a 
simple pointer, like a clock hand, or a compound micro- 
scope, furnished with wires movable by a fine threaded 
screw, or a vernier, so called from the name of the in- 
ventor. 

§ 55. The vernier used in astronomical observations 
is a small arc of a circle, graduated so as not to correspond 
with the graduation of the circle, with which it is to be 
used. For instance, let the large circle be divided into 
tenths of inches, eleven tenths of an inch are on the ver- 
nier divided into ten equal parts. Each division of the 
vernier then contains ^ of each division of the circle. 

When applied to the circle, the first division enables you 
to measure T ^ of "an inch, the second division T §^, and 
so on. This is clone with much more accuracy than if 
each tenth of an inch were divided into ten equal parts. 
Besides only a small arc need be divided, and this can be 
screwed on wherever it is required. Double verniers and 
even those more highly divided are used. They are read 
off by microscopes. 

§ 56, It is no easy thing to divide the circumference 
of a circle accurately into 360 equal parts, and these 
again into smaller subdivisions. An angle of one minute 
occupies on the circumference of a circle of ten inches in 
radius only about ¥ |ty P ar ^ °f an inch, a quantity too 
small to be certainly dealt with without the use of magni- 
fying glasses ; yet one minute is a gross quantity" in the 
astronomical measurement of an angle. With the instru- 
ments now employed in observations, a single second is 
rendered a distinctly visible and appreciable quantity. 
Now the arc of a circle subtended by one second is less 
than the 200,000th part of the radius, so that on a circle 
of six feet in diameter it would occupy no greater linear 
extent than -5-fW part of an inch ; a quantity requiring a 
powerful microscope to be discerned at all. Modern ar- 
tists, however, carry these divisions to great delicacy. A 
circle of three and one quarter feet in diameter may now 
be divided into 10,800 equal parts ; and each of these by 
its accompanying micrometrical apparatus into 1,200 subor- 
dinate intervals. 



ELEMENTS OP ASTRONOMY. 39 

§ 57. A large graduated circle, such as has been de- 
scribed, is sometimes supported in the plane of the meri- 
dian, on a long and powerful axis, and this axis is let into a 
massive wall or pillar. It is hence called a mural circle. 
The meridian being at right angles to all the diurnal cir- 
cles described by the stars, its arc intercepted between any 
two of them will measure the least distance between them, 
and will be equal to the difference between their declina- 
tions, or to the difference between their meridian altitudes. 
These differences are then the angular intervals directly 
measured by the mural circle. But from these it is easy 
to conclude not only their differences but the altitudes and 
the declinations themselves ; for the declination of a body 
is the complement of its distance from the pole. The pole 
being a point in the meridian might be directly observed 
on the limb of the circle, if any star stood exactly therein ; 
and thence the polar distances and the declinations of all 
the other stars might be at once determined. But this 
not being the case, a bright star near the pole is selected 
and observed in its upper and lower culminations ; that is 
when it passes the meridian above and below the pole. 
Now as its distance from the pole remains the same, the 
included arc equals twice the polar distance of the star. 
The polar point being known, the polar distances become 
also known. 

§ 58. The polar star, which is very brilliant, and only 
one and a half degrees from the pole, is usually chosen for 
this purpose. Both its culminations taking place at great 
and not very different altitudes, the refractions are nearly 
equal. Its brightness also allows it to be easily observed 
in the day time. This star is useful for the adjustment 
and verification of instruments of almost every description. 
In the case of a transit, it furnishes a ready means of as- 
certaining whether the plane of the telescope's motion co- 
incides with the meridian. For since this latter plane 
bisects its diurnal circle, the eastern and western portion 
of it require equal times for their description. If, there- 
fore, the upper and lower transits follow at equal intervals 
of twelve sidereal hours, we may conclude that the plane 
of the telescope's motion is in the meridian. 



40 ELEMENTS OF ASTRONOMY. 

The place of the polar point on the limb of the mural 
circle once determined, becomes an origin, or zero point, 
from which the polar distances of all objects referred to 
other points on the same circle are reckoned. 

§ 59. A point on the limb of the mural circle, not less 
important than the polar point, is the horizontal point, 
which, being once known, becomes in like manner an ori- 
gin or zero point, from which altitudes are reckoned. The 
principle of its determination is nearly the same with that 
of the polar point. Two points are to be found on the 
limb, one of which shall be as far below the celestial hori- 
zon as the other is above it. For this purpose a star is 
observed at its culmination, by pointing the telescope di- 
rectly to it, and again by pointing to the image of the same 
star reflected in the still, unruffled surface of a fluid in per- 
fect rest. The image is as much depressed beneath the 
horizon as the star is elevated above it. The point of bi- 
section of the arc which measures their distance is the 
horizontal point. 

§ 60. A divided circle is sometimes permanently fast- 
ened at the axis of a transit instrument, the reading being 
performed by microscopes fixed on the piers. It serves 
for the simultaneous determination of the right ascensions 
and polar distances of objects observed ; the time of transit 
being noted by the clock, and the circle being read off by 
the microscope. This is called the transit circle. 

§ 61. The transit and mural circle are essentially me- 
ridian instruments. But we should possess the means of 
observing an object not only on the meridian, but at any 
point in its course, or wherever it may present itself in the 
heavens. Now a point in the sphere is determined by 
reference to two great circles, one of which passes through 
the pole of the other. On the earth the position of a place 
is known if we know its longitude and latitude ; in the 
heavens if we know its right ascension and declination ; in 
the visible hemisphere if we know its altitude and azimuth. 

To observe an object at any one point we must be able 
to direct the telescope to it. The telescope must therefore 
be capable of motion in the planes at right angles to each 
other j and the amount of its angular motion in each must 



ELEMENTS OF ASTROXOMY. 41 

be measured on two circles, whose planes must be parallel 
to those in which the telescope moves. This is effected by 
making the axis of one of the circles penetrate that of the 
other at right angles. 

§ 62. There are but two positions in which such an 
apparatus can be mounted so as to be of any practical util- 
ity in astronomy. The first is when the principal axis is 
parallel to the earth's axis, and therefore points to the 
poles of the heavens which are the vanishing points of all 
one set of parallels ; and the perpendicular to this axis 
circle has the equinoctial for its vanishing circle, and meas- 
ures by its arcs read off hour angles or differences in right 
ascension. In this position the apparatus is called an 
equatorial. It is one of the most convenient instruments 
for all such observations as require an object to be kept 
long in view, because being once set upon the object, we 
can follow it as long as we please by a single motion, by 
turning it round on its polar axis. In many observa- 
tions this is an inestimable advantage. To counteract the 
apparent diurnal motion of the celestial objects, which is 
continually throwing them out of the field of ordinary tele- 
scopes, (a great annoyance, especially when high powers 
are employed,) a clock-work is attached to the equato- 
rial axis, so constructed as to give to the instrument a quiet 
and steady sidereal motion, contrary to the motion of the 
earth, and which by a slight modification may be applied 
to the solar or lunar motion ; but it is generally sufficient 
when adjusted to a star. The effect of this arrangement 
is to keep the object for several hours constantly in the 
centre of the field of view. 

§ 63. The other position for such an apparatus is that 
in which the principal axis occupies a vertical position, and 
one circle corresponds to the celestial horizon, and the 
other to a vertical circle of the heavens. 

The angles measured on the former are azimuths or dif- 
ferences in azimuth, and those on the latter zenith distances 
or altitudes, according as the graduation commences from 
the upper point of its limb, or from a point 90° distant from 
it. The vertical position of its vertical circle may be known 
by a plumb line, which, however the circle be turned round. 



42 ELEMENTS OF ASTRONOMY. 

should always intersect a mark placed near its lower ex- 
tremity. The north or south point on the horizontal circle, 
is ascertained by bringing the vertical circle to coincide 
with the plane of the meridian. If the zero on the horizontal 
circle is brought into the plane of the meridian, and the 
telescope is turned to a star east or west of that circle, the 
azimuth of the star may be read off the horizontal circle at 
once. 

§ 64. The north or south point on the horizontal circle 
may likewise be ascertained by the method of equal alti- 
tudes. Let a bright star be observed at some distance 
east of the meridian, by bringing it on the cross wires of 
the telescope. In this position let the horizontal circle be 
read off, and the telescope securely clamped on the vertical 
one. When the star has passed the meridian, and is de- 
scending, let it be followed by moving the whole instru- 
ment round to the west, without however unclamping the 
altitude circle until it comes into the field of view, and un- 
til, by continuing the horizontal motion, the star and the 
cross of the wires come once more to coincide. In this po- 
sition it is evident the star must have the same altitude 
above the western horizon which it had when first observed 
above the eastern. At this point let the motion be arrest- 
ed and the horizontal circle be again read off. The differ- 
ence of the readings of the horizontal circle will be the arc 
of azimuth described in the interval, and the north or south 
point of the horizon will bisect this arc. 

An altitude and azimuth circle is particularly useful in 
investigating the amount and laws of refraction. The 
paths of stars can be directly traced, and the exact form 
learned into which their orbits are distorted by refraction. 

§ 65. The theodolite is a modification of the altitude 
and azimuth instrument. It is devoted to measuring hori- 
zontal angles between terrestrial objects. 

The sextant is an instrument of great service in nautical 
astronomy. Its construction is simple, consisting of a 
graduated arc of 60°, a small telescope, a movable arm 
with a vernier attached, and two plane mirrors, the one 
fixed and the other moving with the vernier. It is used 
in measuring the angular distances of objeets from each 



ELEMENTS OE ASTRONOMY. 43 

other ; as of the sun from the horizon, or of the moon 
from neighboring stars. This little instrument is justly 
regarded as one of the most useful inventions of modern 
times, since it is only under its guidance that distant voy- 
ages can be successfully accomplished. 

§ 66. Astronomical telescopes have usually a small 
telescope, called a finder, attached to them, with a mag- 
nifying power of not more than ten, and a proportionally 
large field of view. In its focus two wires cross each other 
at right angles. When the finder is properly adjusted, an 
object which has been brought to the intersection of the 
wires, will be seen in the centre of the field of view of the 
large telescope. 

§ 67. The most important astronomical instruments 
have now been described ; it remains to say a little about 
the qualities needed in an observer. Great manual dex- 
terity, accuracy, promptness, and judgment, are required 
merely to make the instrument perform all of which it is 
capable. The least inexpertness, defective vision, slowness 
in seizing the exact instant of the occurrence of a phenome- 
non, or precipitancy in anticipating it, any one of these 
destroys the value of observations. The constant care and 
vigilance of the practical astronomer must therefore be 
directed to the detection and compensation of errors. He 
cannot get rid of them, he must therefore allow for them. 

A curious fact connected with observation has been 
lately recognized. No two observers seem to agree pre- 
cisely in noting the exact instant at which a star crosses 
the spider's line of a micrometer. Hence arises what has 
been termed a personal equation or correction, which must 
be taken into account when the observations of different 
observers are involved in the same calculation. There is 
the same difficulty in fixing the exact distance and relative 
position of two stars, and this cause of error acts differently 
with different observers, and in different angles of position. 
Apparently our judgment of parallelism is greatly affected 
by our attitude, and especially by the difference between 
looking up and looking down. 

§ 68. A difficulty in observation of small objects arises 
from the great light spread around by the brighter stars. 



44 ELEMENTS OF ASTRONOMY. 

Their presence in the field of view is announced by a dawn 
like that of morning, and the astronomer who would keep 
his sight so delicate as to perceive small bodies must pro- 
tect his eyes from this light. For this purpose the larger 
star is sometimes hidden by a fine needle introduced into 
the focus of the eye-piece of the micrometer. The needle, 
which looks like a black bar, is so placed that the star 
comes into the field of view behind it, and leaves the eye 
undazzled and able to observe small objects in the vicinity. 
The large fixed stars, though they have no discs, require 
as broad a bar to hide them as a planet, owing to their ex- 
treme brilliancy. 

Except in that part of the sky which is very near the 
sun even moderately bright stars may be seen at noon-day 
through the telescopes. Yery bright stars may even be 
discerned without a glass from the bottom of a well or 
through the shaft of a tall chimney or a mine. 

§ 69. I cannot better close this account of the obsta- 
cles which practical astronomers encounter, than by a de- 
scription of the manner in which they have been triumph- 
antly overcome by Lord Rosse in the construction of his 
mammoth telescope. Lord Rosse commenced his labors 
twenty years ago. He began by attempting the improve- 
ment of the refracting telescope, but soon gave preference 
to the reflector. He endeavored to produce the true para- 
bolic speculum which should be free from aberration. The 
exceeding delicacy needful in producing this form with 
mathematical accuracy may be judged from the fact, that 
if two specula of six feet in diameter, the one spherical and 
the other parabolic, were pressed into contact at the cen- 
tre, the edges would not diverge from each other more 
than the thousandth part of an inch. He invented a grind- 
ing and polishing machine, and after repeated trials ob- 
tained the means of furnishing specula from one to six feet 
in diameter. 

§ 70. A difficulty no less formidable impeded his ope- 
rations, — the casting a speculum of sufficient size and 
strength. After repeated trials he made one of three feet 
in diameter, cast in sixteen separate portions. By the ex- 
perience he acquired in making this he became acquainted 



ELEMENTS OE ASTRONOMY. 45 

with the method of casting a large speculum in a single 
piece. Several tormenting difficulties attended his first 
efforts. Small air-holes were formed in the metal, and the 
speculum cracked in cooling. A mould of sand, and sub- 
sequently a mould of cast-iron, failed in giving freedom 
from pores. The desideratum was a kind of mould which 
should retain the molten metal, and yet allow the air glob- 
ules to escape. Such was at length discovered, and 
stamped Lord Rosse's name with celebrity, reducing as it 
does the casting of specula to a certainty. The contriv- 
ance consisted in making the bottom of the mould of layers 
of hoop iron, bound closely together, with the edges upper- 
most. The iron conducted the heat away through the bot- 
tom so as to cool the metal towards the top, while the in- 
terstices between the hoops, though close enough to pre- 
vent the metal from running out, were sufficiently open to 
allow the air to escape. 

The first large speculum thus made in a single piece 
was a round plate of metal, three feet in diameter, nine 
inches thick, and upwards of a ton in weight. In a few 
minutes the metal set in a compact form, and while in- 
tensely hot, was conveyed by a railway to an annealing 
oven, a few feet distant from the foundry. The oven was 
nearly red hot when the speculum was shut up in it, and 
from this temperature it was allowed many weeks to be- 
come gradually cool. It was then ground to the proper 
parabolic curve, and polished. 

§ 71. It was ascertained in the following manner when 
the proper parabolic curve was produced. A high tower 
was erected immediately over the speculum. On a pole at 
the top of the tower, ninety feet distant from the speculum, 
the dial-plate of a watch was placed, forming a small round 
object relieved against the sky. The reflection of the 
watch was seen by an eye-piece at the right focal distance. 
When it became perfectly distinct, the mirror had received 
its proper concavity. It was then placed in its box, lined 
with felt and pitch, so as to prevent any sudden change of 
temperature from greatly affecting its figure. It is singu- 
lar that a nearly similar mode was devised for the bedding 
of specula by Lord Eosse in Ireland and by Sir John Her- 



46 ELEMENTS OF ASTRONOMY. 

schel at the Cape of Good Hope. Sir John Herschel found 
that a speculum supported on three metalic points in the 
circumference made the image of every considerable star 
triangular, and that a packthread stretched down the back 
of the mirror for support distorted the images of the stars 
to a preposterous extent. He employed a great many 
thicknesses of blanket to prevent the effect of flexure in 
the wooden back of the case. To keep the elasticity of 
the fibre the blanket must be often shaken. 

§ 72. The speculum so fortunately completed by Lord 
Rosse was fixed or bedded on three iron plates, which gave 
it support, and then transferred to its appointed situation 
in the tube. This is three feet in diameter, and thirty 
feet long, and attached to an apparatus on the lawn, 
by which it can be brought to bear on any point of the 
sky from a short way above the horizon. The machine- 
ry for moving it round and raising and depressing it 
is simple and ingenious ; and notwithstanding its size, it 
may be adjusted with the greatest ease. Two step-ladders 
form part of the apparatus, and by these we mount to a 
gallery, which can be raised or lowered to any required 
height. In order to procure an observation, the tube is 
first brought to bear on the star or other object, and the 
gallery being raised, we ascend to it by one of the ladders. 
On reaching the gallery, which is a small railed platform 
sufficient to hold several persons, we find ourselves close to 
the telescope, near its upper extremity ; and here, on 
looking through a small eye-piece fixed to the tube, we at 
once recognise, in the obliquely-placed mirror within, the 
object of our observation. The tube is of wood, hooped 
with iron. The mouth of the tube remains permanently 
open. The telescope is lowered in w r et weather, and the 
speculum is confined in a case, the cover of which is with- 
drawn by an exterior action when required. A vessel of 
quick-lime is also kept constantly in the case, for the pur- 
pose of absorbing the moisture and acid vapors, by which 
the speculum might be tarnished. 

§ 73. The eye-pieces used with this telescope range 
from 180 to 2,000 times the power of the naked eye. 
Unless the atmosphere be exceedingly clear, a powerful 



ELEMENTS OF ASTROXOMY. 47 

eye-piece will magnify the globules of watery vapor, and 
form a haze. Different densities from contending streams 
of cold and warm air have a similar effect ; and if the 
atmosphere be exceedingly cold, as in a Russian winter, 
floating spicula of ice, invisible to the naked eye, are mag- 
nified so as to interrupt perfect observation. 

Sir John Herschel found that the excessive heat and 
dryness of the sandy plains at the Cape often destroyed 
distinct vision, and that in a very singular manner. 
In some cases the images of the stars are violently di- 
lated, and converted into nebulous belts or puffs of 10" or 
15" or more in diameter. In others they form soft, quiet; 
round pellets of 3" or 4" in diameter, very unlike the 
spurious discs which they present when best defined, and 
rather resembling planetary nebulas. Sometimes the 
structure as it were of these pellets is disclosed, and they 
are seen to arise from an infinitely rapid vibratory motion 
of the central point in all possible directions, while on a few 
occasions the appearances are exceedingly perplexing and 
singular. Some of the phenomena evidently have refer- 
ence to the state of the air in the tube of the telescope ; 
the tube of a reflector being necessarily open at the mouth, 
ascending and descending currents of hot and cold air, 
usually rotating spirally, are established, and are very 
prejudicial to distinct vision. The remedy is to dispense 
with a tube altogether, substituting for it a light, strong, 
inflexible, iron frame work. In refracting telescopes, 
where the air is completely inclosed, its circulation is not 
nearly so injurious. 

§ 74. The performances of Lord Rosse's telescope 
were found to be far beyond those of any previously con- 
structed instrument. But Lord Rosse considered that 
something still grander could be achieved ; and before the 
above telescope was well finished, he projected one of the 
extraordinary dimensions of six feet diameter in the specu- 
lum, with a tube of sixty feet long. The casting, grind- 
ing, polishing and mounting of this monster speculum were 
pretty nearly a repetition, on a larger scale, of what had 
been previously done. Its focal length is fi%-three feet ; 
it weighs nearly four tons ; and, as its diameter is six feet, 



48 ELEMENTS OF ASTRONOMY. 

it has an area four times greater than that of the three- 
feet speculum* When finished, the speculum was placed 
in a square box, which is attached to the lower end of the 
tube, and by means of a door can be entered at pleasure. 
This box adds six feet to the length of the tube, which, 
like its predecessor, is of wood, hooped with iron like a 
barrel, and so wide that a tall man could walk through it 
without stooping. This huge black funnel is suspended 
between high and strong walls. It swings with a clear 
space of twelve feet on each side ; and so far it can be 
drawn aside, giving half an hour before and after meridian. 
By means of a windlass, and a most skilful adjustment of 
chains and counterpoising weights, it can also be brought 
to the zenith, or turned fairly round from south to north. 
Enormous as are its dimensions, and although weighing 
altogether twelve tons, it seems to be about as easily 
moved as the other telescope ; and it is as much in the 
mechanical contrivances for effecting this purpose as in any- 
thing else that the peculiar merit of the structure consists. 



CHAPTER IV. 

NEBULAR AND SIDEREAL SYSTEMS. 

The Milky way. Comparative dimensions of the Solar and Sidereal Sys- 
tems. Distances of the Fixed Slars. Classification of Stars according 1 
to their apparent Magnitudes. Distribution of the Stars. Gauging of 
the Heavens by Herschel. True form of the Milky Way. Clusters and 
Nebulse. Forms and distribution of Nebulse. Vastness of the Universe. 
Effect of the finite velocity of Light. 

§ 75. In that portion of infinite space which is unveiled 
to the gaze of man lie clusters of countless suns, separated 
one from another by unimaginable intervals. Within one 
of these clusters, and probably nowise distinguished as to 
size or brightness from the other orbs, lies the sun around 
which our earth revolves. About this sun, at distances 
too small to be represented here, revolve the members of 



ELEMENTS OF ASTRONOMY. 49 

the solar system. With, this bright company are we envi- 
roned day and night. 

* Fig. 1, plate I, represents the outline of a section of the 
cluster to which it is supposed our sun belongs. The sec- 
tion makes an angle of 35 degrees with the earth's equator, 
crossing it in 124J° and 304^° of right ascension. A ce- 
lestial globe adjusted to the latitude of 55° north, and 
having a Ceti near the meridian, has the plane of this sec- 
tion pointed out by the horizon. It cuts the milky way at 
right angles on one side in its two branches which cross the 
constellation of the Eagle, and on the opposite side in the 
southern part of the Unicorn towards the Canis Major. The 
circle in the figure includes all the suns or stars ever visible 
to the naked eye. On all sides of the earth, taken to- 
gether, from four to eight thousand (for the number is dif- 
ferently estimated) may be seen. About two thousand may 
be seen by average eyes on an ordinary night in clear cli- 
mates. In foggy island climates not more than nine hun- 
dred are visible at once. Inexperienced observers suppose 
the number much larger, partly because the sight is daz- 
zled by their irregular distribution, and partly because as 
they diminish from stars of large size to those scarcely visible 
the imagination supposes others still smaller and invisible. 
The dot in the centre of the figure gives the position of our 
sun in the cluster. All stars beyond the circle, if they lie 
scattered in space are invisible to the unassisted eyes of the 
inhabitants of the earth. If they lie many in one direction; 
they present to the naked eye a milky, hazy appearance, 
and are called by the general name of nebulae. This 
nebulous light belongs to distant groups of our cluster, and 
also to more remote clusters. Stars one hundred and 
eighty times the distance of Sirius are the most remote 
which appear even as nebulae to the naked eye. 

§ 76. In order better to conceive of the dimensions of 
the solar system, and of our cluster, and the intervals 
which lie between the clusters, let us make ourselves famil- 
iar with known and moderate distances, and advance from 
these to those which almost baffle the imagination. 

The sun is a globe 383,000 miles in diameter. A hol- 
low sphere with a radius of three thousand millions of miles 
5 



50 ELEMENTS OF ASTRONOMY. 

includes all the members of the solar system yet discov- 
ered. If the swiftest race horse had begun to traverse 
this sphere at full speed at the birth of Moses, thirty-four 
centuries ago, he would not yet have accomplished one 
quarter part of his journey. 

§ 77. Between the solar system and the stars lies 
a wide space traversed only by comets, and their appro- 
priate field, if indeed they be not visitants from other 
spheres. The nearest fixed star, u, Centauri, is twenty-one 
millions of millions of miles from the sun ; 61 Cygni is 
fifty-six millions of millions distant from it. The distances 
of but few of the fixed stars have yet been ascertained. 
What we know of their distribution makes it probable that 
the stars of one cluster are on an average separated among 
themselves by distances as great as that between our sun 
and the nearest fixed stars. 

The distances of those stars which have not yet been 
measured can only be inferred from their superior bright- 
ness. And here again for w T ant of knowledge we must in- 
troduce another supposition. We must suppose that stars 
appear large merely in consequence of their proximity to 
us ; and we must leave out of sight the differences which 
have lately been proved to exist in their actual size, or in 
the intrinsic brightness of their surfaces. As we know r 
nothing of these particulars, and as they may vary in dif- 
ferent stars in the ratio of many millions to one, w T e cannot 
be sure that we assign to any star its true distance. An 
arrangement of the stars in the order of their precise ap- 
parent brightness is much to be desired ; but the variety of 
their color makes such an arrangement difficult. If they 
were catalogued according to the force of the whole im- 
pression made on the eye, we might obtain some knowledge 
about their intrinsic light-giving power, and might ascer- 
tain the extent of the changes which take place in the light 
of some of them. 

§ 78. At present the stars are loosely divided into 
classes according to their apparent size. All above a cer- 
tain size are considered of the first magnitude ; all less 
bright than these, and above a certain brightness, are of 
the second magnitude. Those decidedly inferior in bril- 



ELEMENTS OF ASTRONOMY. 51 

liancy form the third class, and so on down to the sixth 
and seventh magnitudes, which comprise the smallest stars 
visible to the naked eve in the clearest night. Beyond 
these the telescope reveals new orders, and as higher 
space-penetrating powers are used, new orders are added. 
Of course the layer in which a star first appears does not 
give us its position in space, it may be a very large star 
and lie farther off, or it may be a small one and lie in the 
layer in which it first appears to us. All we expect to 
learn is the comparative brightness of the stars as seen 
from the solar system. 

The division into magnitudes is arbitrary, nor is it easy 
to determine where one magnitude ends and another be- 
gins, since all those stars which are included in one magni- 
tude are by no means of the same size. 

The light of Sirius, the brightest of the fixed stars, is 
about 324 times that of an average star of the sixth mag- 
nitude. As might be expected, the number of stars of 
each magnitude increases rapidly as we pass from the 
first to the lower magnitudes. There are from fifteen to 
twenty stars in the first class, and, unfortunately for us 
in the northern latitudes, the largest and most brilliant of 
these are not visible in our heavens. Of the second mag- 
nitudes there are fifty or sixty stars ; of the third, 200 ; 
and in the first seven classes taken together, there are up- 
wards of 2,000 in the northern hemisphere ; in the milky 
way about 18,000,000 ; and in all the nebulas, about 
100,000,000 distinct stars are within reach of telescopic 
vision. 

§ 79. The three or four brighter classes are distrib- 
uted with tolerable equality throughout the heavens, but 
the smaller ones visible to the naked eye increase rapidly 
in number as we approach the borders of the milky way. 
The telescopic stars are crowded beyond imagination along 
that circle and the branch which it sends off, so that its 
whole light is composed of stars, whose average magni- 
tude is not above the eleventh or twelfth. It was com- 
puted by Herschel that in one hour 50,000 passed through 
the field of his telescope in a zone 2° in breadth. This 
compression was partly owing to the vast numbers brought 



52 ELEMENTS OF ASTRONOMY. 

within his line of vision in depth, and partly to the real 
crowding of the stars in the milky way. 

This unequal distribution of stars enables us to learn 
approximately the form of our cluster. Sir William Her- 
schel determined it on the following principle. If you 
were in a crowd or immense building filled with people, 
you would judge your distance from the edges of the crowd 
or from the walls of the hall from the number of people 
seen in each direction. If you were in a wood and wished 
to determine its outline without leaving its interior, you 
might form a rude approximation to the true outline by 
taking your position in one spot and drawing imaginary 
lines in every direction to the edge of the wood. If one 
hundred trees were visible in one direction, you might as- 
sume for the line running thither a certain length, and pro- 
portion all your other lines to this, making them longer or 
shorter as more or fewer trees were visible in each direc- 
tion. A bounding line passing through the termination of 
each of these lines would be not far from the true outline. 
A body which has extension in three directions, as our 
cluster, may be treated in the same manner as the wood. 
Herschel used the telescope as a sounding line, and in- 
ferred from its discoveries the hollows, protuberances, and 
in fact the shape of a great portion of our cluster. He 
made 700 observations to fix its form and dimensions. 

In order to determine the comparative mean richness in 
stars of any two regions of the firmament, Herschel made 
use of a telescope which magnified 187 times, and whose 
field embraced a circle of 15' diameter. This field in- 
cluded each time about the eight hundred and thirty thou- 
sandth part of the entire heavens. Towards the middle of 
the first of these regions he counted successively the num- 
ber of stars included in ten fields contiguous or at least 
very near each other. He added these numbers and di- 
vided the sum by 10. The quotient was the mean richness 
of the region explored. The same operation, the same 
numerical calculation, gave him the mean richness of the 
second region. When the latter result was double, triple, or 
tenfold the former, he inferred that a stratum of it contain- 
ed twice, three times ? or tenfold as many stars as a stratum 



ELEMENTS OF ASTRONOMY. 53 

of the former of equal depth, and consequently that the 
stars extended twice, three times, or ten fold as far in the 
latter as in the former direction. In some portions of the 
sky at least four successive fields were required to meet 
with three stars. Elsewhere these circular areas of 15' 
diameter contained 300, 500, 588 stars. 

To sound the whole heavens in this way would require 
more than a million of observations. Herschel made 700 
soundings, omitting however the circumpolar regions. He 
did not therefore learn with certainty the whole form of our 
cluster. 

Suppose the whole sphere, as far as the eye can reach, 
to be represented by a common two-feet celestial globe. 
Herschel sounded what would be in proportion to this globe 
one hundred and fifty feet beyond its centre, and made a 
chart of it in section extending to a proportional distance 
from our sun. On the same scale, a chart of the discov- 
eries possible to Lord Rosse's telescope, would have for 
the radius of the sphere from which it was taken one thou- 
sand feet. Herschel's observations, though taken from the 
earth as a centre, apply equally to the sun as a centre, for 
the ninety-five millions of miles which separate the sun and 
the earth make no more difference in the position of those 
bodies than the distance between two observers who side 
by side watch the setting sun makes in the position of that 
luminary. 

§ 81. Studying thus the heavens, Herschel found on 
most sides only a small number of large stars ; but toward 
the bright belt called the milky-way, are myriads of stars, 
so distant as only to be visible by a faint white light. 

The increase of numbers in approaching the milky-way 
is imperceptible among stars of a higher magnitude than 
the eighth, and except on the verge of the milky-way 
itself, stars of the eighth magnitude can hardly be said 
to participate in the general law of increase. For the 
ninth and tenth, the increase, though unequivocally indi- 
cated over a zone, extending at least 30° each side of 
the milky-way, is by no means striking. It is with the 
eleventh magnitude that it first becomes conspicuous, 
though still of small amount when compared with that 
5* 



54 ELEMENTS OF ASTRONOMY. 

which prevails among the mass of stars of magnitudes in- 
ferior to the eleventh, which constitute sixteen-seventeenths 
of the stars within 30° on each side of an imaginary circle 
running through the middle of the milky-way. Two con- 
clusions follow from this ; first, that the larger stars are 
realty nearer to us than the small ones ; secondly, that our 
system is plunged in ihe sidereal stratum constituting the 
galaxy, to a depth equal to about that distance, which 
corresponds to the light of a star of the ninth or tenth mag- 
nitude, and certainly does not exceed that corresponding to 
the eleventh. 

Applying to this cluster the principle by which the wood 
was mapped, it must be by no means globular, but rather 
like a slice through the centre of a globe, extending far- 
thest in the direction of the milky-way. Its major axis is 
estimated at seven or eight hundred, and its minor axis at 
a hundred and fifty times, the distance of Sirius. It is 
however by no means certain that Herschel sounded to 
the limits of our cluster. 

§ 82. Oar cluster has been compared in form to a 
grindstone, a little bulging instead of plane on the sides, 
and having its rim split through about one third of its cir- 
cumference, but not nearly to the centre, the parts divid- 
ing at an angle of about 30°. A section of this at right 
angles to the plane of the circumference would have nearly 
the form of the letter Y. Our sun is just below the cleft 
or the joining of the Y, near the vertex of the angle, nearer 
to Sirius than to the Eagle, nearer to the Southern Cross 
than to Cassiopeia, and almost in the middle of the starry 
stratum in the direction of its thickness. 

Suppose the grindstone to be very porous. Let its mi- 
nute atoms represent stars, the pores between being the 
interstellar spaces. An observer within such a cluster 
would have a scene resembling our own celestial vault. 
Toward the sides comparatively few stars would be seen ; 
toward the circumference a succession of remote stars 
would form a zone lying like our milky-way. Thus on our 
blank sides only forty stars in succession are seen, in the 
direction of the milky-way nine hundred may be counted, 
though this is not in all parts equally profound. 



ELEMENTS OF ASTRONOMY. 55 

Between it and the central portion of our cluster are 
breaks and vacuities, which detach it and make it appear 
more like a separate ring of stars. The ring in many parts 
consists of separate groups of stars, mostly spherical in 
form, the groups sometimes lying close together, sometimes 
having spaces between. In some parts there is an average 
of 3,138 stars in a square degree, and in the denser part 
5,093 in the same area. It varies in breadth from 5° to 
10° and even 16°. Its telescopic breadth is 6° or 7° 
greater than that assigned by the naked eye, the stars be- 
coming much less numerous near its edge. Even the 
apparent breadth of 5°, in its most distant parts of 900 
units of distances of Sirius gives a real linear breadth of 
78 units of distances of Sirius. Therefore the milky way 
greatly exceeds even in breadth the reach of the naked 
eye. It twice separates and unites. Its cleft part ex- 
tends from below the ecliptic in the southern hemisphere 
to very near the pole of the ecliptic in the northern hemi- 
sphere. 

§ 83. Its two branches between Serpentarius and An- 
tinous separate more than 22° of the sphere. It is in- 
clined to the earth's orbit about 60°, and its edge is not 
more than 22° distant from the poles of the ecliptic. In 
northern latitudes it is most conspicuous from July to No- 
vember. It is visible by night at all seasons, in all posi- 
tions of the globe. As seen by us projected on the celes- 
tial sphere it differs but a few degrees from a great circle, 
we being almost in its plane. If we were more on one 
side of its plane it would appear like a small circle, and 
would divide the heavens into two quite unequal parts. 
This gives us our position as to one dimension of our 
cluster. 

"We learn from other appearances that our north pole is 
farther from the milky way than our south pole ; that is, 
that we are nearer that part of the ring which approaches 
our south pole than that which approaches our north pole. 
Not only does the telescope give more soundings toward 
that part of the ring which encircles our northern hemi- 
sphere, bat the whole light in the northern hemisphere is 
fainter, and evidently comes from exceedingly remote 



56 ELEMENTS OF ASTRONOMY. 

stars. In the southern hemisphere not only is the belt a 
blaze of light, but many more stars of the first magnitude are 
discernible, proving our greater proximity to them. From 
Sirius to Antinous it is perfectly illuminated with stars, 
many of which are visible to the naked eye, while toward 
the north the light gradually becomes hazy and without a 
trace of stars. The very great size of the southern stars 
makes it probable that we lie not very far inside the de- 
tached ring. 

§ 84. As from the contemplation of the solar system 
we stretched our imagination to embrace the fixed stars, 
we must now pass on and conceive these myriads of stars 
forming a cluster, isolated in space, and separated from 
other clusters by intervals in most cases far greater than 
the distance between our sun and the most remote portion 
of the milky way. These clusters seen through the tele- 
scope vary in size, brightness, number, and color of stars, 
and present a variety of fanciful shapes. The elliptic 
form however prevails with an increased brightness in the 
centre greater than would arise merely from the depth of 
the cluster, and which must be attributed to the greater 
proximity of the central stars owing to their mutual attrac- 
tion. The stars which compose the clusters also show the 
influence of gravity by their disposition to break into 
groups. 

The nearer clusters offer to the eye only a faint diffused 
light. Viewed through a telescope of moderate power, 
they resemble a handful of fine sparkling sand, or, as it is 
called, star dust. A higher power brings distinct stars 
into view at small intervals, and with a faint light. The 
stars near the edges may be distinctly seen, while those 
near the centre unite their rays and form a brilliant light. 
A higher power enables us to see individual stars with 
great distinctness. Many clusters contain ten or twenty 
thousand stars wedged together in a space whose area is 
not more than one tenth part of that of the moon, so that 
the centre where the stars are seen one behind the other 
is a blaze of light. 

Herschel applied to these clusters the principle by which 
he had ascertained the distance of the stars of one cluster. 



ELEMENTS OE ASTRONOMY. 57 

He thus ascertained that the distances of forty-seven re- 
solvable clusters were at least 900 times that of Sirius 
from the sun. A cluster first resolved by Lord Rosse's 
mirror must lie six thousand times as remote from us as 
the nearest fixed stars. 

§ 85. Before this immense power was turned upon the 
nebulae it was supposed that besides those clusters of stars, 
which from their distance take a nebulous appearance, 
there was floating in the sphere a very thin nebulous sub- 
stance, perhaps the material from which stars were con- 
densed. The immense size of some nebulise seemed to 
make it improbable that they could consist of clusters of 
stars. Lord Rosse's telescope has, however, resolved 
some of the largest and faintest of these nebula, and thus 
made it probable that all of them may yield to yet more 
powerful instruments. The distinction, therefore, between 
nebulas and clusters does not properly exist in nature, but 
probably arises from the low power of the instruments 
used. When the stars which compose a cluster are so 
small, or so close, or both, as not to be separable by the 
telescope, the cluster may offer every variety of illumina- 
tion from a mere vaporous patch of light to a brilliant sur- 
face of mottled or even of sensibly uniform illumination. 
In this case, if the nebula is very distant, it may present 
a uniform disc like a planet, more or less well defined and 
uniform, and will be called a planetary nebula. These 
objects are seen in both hemispheres, but less so in the 
southern than in the northern. If, however, the nebula 
be not only globular, but compressed towards the centre s 
when exceedingly remote it will have the appearance of a 
single star surrounded by nebulous matter. Sometimes 
nebulous matter appears appended to a nucleus like that 
of a comet. In this case the cluster is irregular and 
the condensation not central. For the present, however, 
the division of nebulas into resolvable (clusters) and irre- 
solvable, facilitates their description. 

§ 86. Nebulas are also divided as to their form into 
regular and irregular ; the regular form being elliptic, with 
a diminution in the ellipticity of their strata from without 
inwards, so as to approach a spherical nucleus, however 



§8 ELEMENTS OF ASTRONOMY. 

elongated may be their outline. The regular nebulae ap- 
pear globular to all but the most powerful instruments, be- 
cause the outer layers are usually faint. From this cause 
it is generally asserted that the globular form prevails. 
Annular nebulae are considered regular, for they are prob- 
ably hollow spherical shells of stars. Planetary nebulae 
and globular clusters also come under this head. 

Irregular nebulae are the most extensive objects in the 
heavens. Their forms are most capricious, imitating a 
dumb-bell, a fan, and in one instance a human head and 
breast. Their true form is not however seen by us, and it 
may be that many nebulae lying nearly in our visual line are 
by our eye blended into one. Probably in some cases they 
really consist of systems of systems. Indeed double nebulae, 
or those lying very near one another, occur so frequently that 
their proximity cannot be attributed to chance. The nebu- 
lous system appears to be distant from the sidereal, though 
involving it, and perhaps to our eyes intermixed with it. 
The limits of our sidereal system have not in all directions 
been defined. The distribution of the nebulae is not like 
that of the milky way in a zone or baridv- One third of 
the nebulous contents of the heavens are congregated in a 
broad irregular patch occupying about one eighth of the 
whole surface of the sphere, almost entirely situated in 
the northern hemisphere, and occupying the head and 
shoulders of the Virgin, and the surrounding constellations. 
Within this region are several local centres of accumula- 
tion where the nebulae are exceedingly crowded. The 
lesser nebulous region of the northern hemisphere lies in 
Pisces, and is much less concentrated. 

§ 87, In the southern hemisphere a more uniform dis- 
tribution prevails. With the exception of the two Nubecu- 
lae, which are full of nebulae, and the greater of which is 
even richer than the denser portion of the northern group, 
this hemisphere contains alternating patches of nebulae and 
vacuities of greater or less extent. In one of the vacuities 
in which comparatively few nebulae occur the south pole is 
situated, having one nebula, however, within half a degree 
of it, as the north pole also has one within five or six min- 
utes. This barren region extends nearly 15° on all sides 



ELEMENTS OF ASTRONOMY, 59 

of the pole, and immediately on its borders occurs the 
smaller Nubecula. One of the most remarkable features 
in the southern nebulous system is the extraordinary dis- 
play of fine resolvable and globular clusters which occur 
in the region occupied by Corona Australis, the body and 
head of Sagittarius, the tail of Scorpio, with part of Tele- 
scopium and Ara. Here in a circular space of 18° in ra- 
dius are collected no less than thirty of these beautiful ob- 
jects. Are we to suppose them to be a bunch of general 
nebulous systems nearer to us than the rest ? Or is it 
merely that on this side we approach nearer the milky 
way ? It cannot be doubted that some of these objects 
form a part of the milky way. 

§ 88. The bright fleecy spots, long known to mariners 
as the Magellanic clouds, are composed of large patches of 
unresolvable nebulas, and of nebulosity in every stage of 
resolution, up to perfectly resolved stars like the milky 
way, as also of regular and irregular nebulae, properly so 
called, of globular clusters in every stage of resolvability, 
and of clustering groups sufficiently insulated and con- 
densed to come under the denomination of clusters of stars. 
The Nubecula Minor contains within an area not much ex- 
ceeding ten square degrees, forty-three nebulae and clus- 
ters ; the Nubecula Major, within an area of about forty- 
two square degrees, contains 278, without reckoning fifty 
or sixty outlines, making an average of about six and a 
half to the square degree, which very far exceeds any 
thing that is to be met with in any other regions of the 
heavens. This intermixture of stars and unresolved nebu- 
losity makes it probable that the nubiculae are systems 
which resemble none in our hemisphere. Sir John Her- 
schel has ascertained the places of 919 stars, nebulae and 
clusters in the Nubecula Major, and of 244 in the Nubecu- 
la Minor, as an approximation toward a catalogue of the 
objects they contain. He has also fixed the places of 
4,015 nebulae or clusters, of which the southern hemi- 
sphere contains the larger portion. Each hemisphere con- 
tains about as many as the eye sees stars on an average 
night. 

Numerous and vast as these clusters are, the distances 



60 ELEMENTS OF ASTRONOMY. 

which separate them are yet more astounding. Imagine 
clusters of suns, each sun lying so far distant from the 
other that the eye can pass over only six such intervals in 
one direction, while the cluster contains from end to end 
hundreds of such suns lying at such intervals. Then im- 
agine these clusters lying so widely separated from one 
another that they are but as handfuls of dust in space. 
How wide must be that universe of which man cannot com- 
prehend a corner ! 

§ 89. It may assist us to realize their vast distances if 
we consider how long light travelling 192,000 miles a second 
would be in travelling from them to us. Light is one and 
one quarter seconds passing from the moon to the earth ; 
eight minutes from the sun ; three to twelve years from 
the nearest fixed stars ; 140 years from the most distant 
stars visible to the naked eye ; thousands of years travers- 
ing our cluster in its longest direction, from Aquila to 
Monoceros ; and millions of years coming from distant 
clusters, a period long enough to allow important changes 
in the cluster from which it emanates. Thus the moon 
may have been dispersed into atoms for more than a sec- 
ond, and the sun for eight minutes, and we should still see 
them perfect and entire. The star a. Centauri may have 
changed its color three years ago, and we should still see 
it of its former hue. The bright star Vega, must have 
been placed in the heavens nine years before its rays 
struggled to our little world ; and more distant stars may 
have been shining for centuries yet not so long that their 
light has reached the earth. The light which now meets 
our eyes may come from stars long since quenched in dark- 
ness. A human being may be born, pass through the 
seven stages of life and die, while light from the smallest 
stars visible to the naked eye is reaching us. Nay, our 
whole historic period is about the length of time which 
light occupies coming from the nearer or cleft edge of our 
cluster to the earth. Thus the astronomer who records 
the aspect and variations of a distant nebulae gives its his- 
tory millions of years since. If the solar system and the 
fixed stars were called into existence at the same moment, 
from the earth no other body would at first have been seen. 



ELEMENTS OE ASTRONOMY. 61 

The moon would have appeared in a second and a quarter, 
the sun in eight minutes, the stars would have peeped out 
one by one in the course of years ; there would have been 
no field for the telescope under a century. An exact 
chronicle of their times of appearing would have been a 
perfect measure of their respective distances from us. 



CHAPTER V. 

INTERIOR OF OUR SIDERAL SYSTEM. 

Absolute and Relative Motion. Motions of the Fixed Stars. Proper Mo- 
tion of 61 Cygni, and of Arcturus. Motion of the Solar System. In- 
vestigations of Herschel, Struve, and Argelander. The Central Sun, 
Double and Multiple stars. New Stars. Variable Stars. Color of 
Stars. 

§ 90. Having obtained a general idea of the form of 
our cluster, and of its distance from other clusters, we 
will now study its interior. We will inquire whether the 
bodies which compose it move among themselves, and 
whether they undergo any changes of constitution ? 

Of absolute motion we can know nothing. Relative mo- 
tion is all it concerns us to know. But we want some 
fixed point to which we may refer motions ; and the heav- 
ens afford us no such fixed point. The cluster of which 
our sun is a unit may, for all we know, be rushing on with 
unimaginable speed ; but its suns retaining their relative 
position would still appear to be at rest. Our sun may be 
changing his place among the stars, but only centuries of 
observation can make his motion evident. The earth is 
certainly revolving round the sun and rotating on its axis. 
These two latter motions we perceive by reference to the 
stars, which do not partake of them, but how are we to 
ascertain any motion we have in common with the stars. 

We know not whether there be in the universe one 
star deserving to be called absolutely fixed, and we 
6 



62 ELEMENTS OF ASTRONOMY. 

must be contented to refer our motion to the bodies 
nearest us. 

§ 91. By observing and reasoning we shall be able to 
ascertain whether the motions discovered are entirely our 
own, entirely belonging to the stars, or compounded of 
both. We have instances of all these kinds of apparent 
motion in the heavens. The rising and setting of the sun 
and its yearly motion are entirely apparent and owing to 
the earth's motion. The moon's motion is made up partly 
of our real motion, partly of her revolution ; and besides 
this the moon and earth share a third motion round the 
sun. We do not see in the solar system a motion which is 
not influenced, either accelerated, delayed, or changed in 
direction by the earth's own motion. When we inquire 
into the motion of the fixed stars, the earth's proper mo- 
tion no longer embarrasses us ; it is too small to be of the 
least account ; but we must allow for possible motion of the 
sun and solar system together. We must inquire whether 
this motion belongs to them, or to us, or to both. We 
must observe whether the apparent motion is common to 
all the stars, whether it is in such a direction that the mo- 
tion of our sun would account for it, or whether absolute 
and parallactic motions unite in producing the apparent 
motions of the stars. 

§ 92. The fact that the fixed stars change place 
among themselves has long been suspected. Maps and 
observations made at intervals of fifty years differ. So 
that it may be affirmed with certainty that a map of the 
heavens, correct this year, will, after a few years, and still 
more after a few centuries, be found faulty. The immense 
distance of the stars causes their motions to appear slight, 
or not to appear at all to the naked eye. Millions of miles 
of their path subtend to our eyes not even an angle of one 
second only. The annual motion of sixty-one Cygni is more 
than a thousand millions of miles, yet we call it a fixed 
star. To the eye of a common observer the heavens pre- 
sent the same features they did thousands of years ago. 
But by comparison of catalogues many minute changes 
may be detected, and modern instruments can measure 
their changes from year to year. 



ELEMENTS OF ASTRONOMY. 63 

As might be expected the larger and nearer stars show 
the most motion. This would be the case whether the 
motion were theirs or ours. The bright star Arcturus is 
moving towards the south-west with a velocity of two 
seconds and a quarter of arc every year. Sirius, Alde- 
baran, Castor, and others, are likewise rapidly moving. 

§ 93. Since so many of the neighboring stars move, 
it is probable that our sun, similar in its nature and subject 
to the same laws, also moves, and that the motions we see 
in the stars are the differences between their motion and 
ours. If the stars are at rest our sun must move. Either 
hypothesis gives the sun proper motion. It remains then 
to ascertain toward what point the sun is moving, and this 
may be done by applying a very simple principle. As w T e 
pass among columns, or the trees of a forest, those we ap- 
proach separate from one another, while those we leave be- 
hind seem to close together. The greatest apparent mo- 
tion is in those stars we are overtaking and leaving behind, 
those which are at right angles to the sun's motion. Her- 
schel judged that the sun was moving towards a point in 
the constellation Hercules, because the distances between 
the stars in that region are becoming greater. This con- 
jecture has lately been confirmed by the investigations of 
Argelander and of Otto Struve. 

§ 94. Argelander compared the positions of 560 stars 
in all accessible regions of the heavens with those laid 
down in the catalogues of preceding astronomers. Of 
these 560 stars, 170 moved so slowly as to yield no relia- 
ble results in the short time between the first and last ob- 
servations. Of the remaining 390, the slowest motion 
gave a yearly change of place amounting to one tenth of 
one second of an arc. These stars were arranged in 
classes according to the rapidity of their motions. The 
first class included all whose yearly motion equalled or 
exceeded one second of an arc, and contained 21 stars. 
The second class included those whose yearly motion ex- 
ceeded one half, and fell short of a whole second. This 
class contained 50 stars. The third class comprehended 
all whose motion was between one half and one tenth of a 
second, and contained 317 stars. 



64 ELEMENTS OF ASTRONOMY. 

§ 95. By a rough examination it became evident that 
the sun belongs to the class of rapidly moving stars, and 
that the solar motion was directed toward some point in the 
constellation Hercules, where indeed the elder Herschel 
had placed it. 

The determination of the exact point was now a matter 
of trial. Argelander examined the motions of the stars of 
the first class. He selected a point in the constellation 
Hercules toward which he supposed the sun moving, and 
computed the direction in which these stars would seem to 
move if their apparent motion were caused by the sun's 
real motion toward the selected point. The computed di- 
rection was compared with the observed direction, and if 
they coincided the computed one would be considered the 
true one. If there were differences, a point must be se- 
lected which would reduce these differences to the least 
possible amount. The three classes of stars were investi- 
gated in this manner, and the point in Hercules, toward 
which the sun was moving in the epoch 1840, was ascer- 
tained. 

§ 98. An attempt has even been made to calculate 
the velocity with which the sun moves in its orbit. The 
star 61 Cygni is known to be 660,000X95,000,000 miles 
from the sun, and its apparent motion through space has 
been accurately determined. Its distance from the point 
of observation being known, that apparent angular motion 
can be converted into an apparent motion of so many miles. 
Now if this displacement of 61 Cygni is owing to the trans- 
lation of the sun, the velocity with which we are darting 
through space may easily be ascertained. 

If so we might move the distance of 61 Cygni in 40,000 
years. In five hundred thousand years we might reach 
the extremest verge at which the eye can descry a single 
star. In two hundred and fifty millions of years we should 
reach the remotest distance to which Lord Rosse's telescope 
can pierce. 

§ 97. But this hypothesis of the sun's motion by no 
means accounts for all the motions observed among the 
stars. The next question is, in obedience to what law the 
sun and stars perform their proper motions. Then again 



ELEMENTS OF ASTRONOMY. 65 

we want long continued observation of the stars, and com- 
parison of catalogues made at great intervals, in order to 
learn the precise direction and quantity of the motion. 
The law which guides the binary systems, and which ap- 
pears to influence even the form of the clusters is the law 
of gravity ; we may therefore suppose it to regulate the 
stars of each cluster. 

Reasoning from analogy with our solar system, we may 
suppose the cluster to contain a great central luminary 
around which all are tending. Reasoning from analogy 
with the multiple stars, we may suppose all to move round 
some central point. 

§ 98. Mr. Maedler, of Dorpat, thinks he has discovered 
this central point. The following is an outline of his rea- 
soning. Since gravitation extends throughout our system, 
there must be a centre of gravity to it. What fills this 
centre, or whether it is filled at all, becomes a matter of 
special research. Admitting the existence of one grand 
central body, visible or invisible, predominant over others, 
we should expect to find the most rapid motions in its im- 
mediate vicinity. That many of these revolving bodies 
and even the central body itself may be invisible, will not 
be denied for a moment ; but as we are able to look out 
upon the starry heavens in every direction, we ought to 
find a point in which the most rapid motions are concen- 
trated, and from this point outwards the quantity of motion 
should continually decrease. That an apparently more 
rapid motion than that found near the centre cannot be 
predicated of any stars except those very near our sun is 
manifest. 

§ 99. If however we adopt the hypothesis of the cen- 
tre of gravity merely, the proper motions of the stars in 
the neighborhood of it will be feeble. In this direction 
rapid proper motions can exist only in those stars in which 
their great proximity to our sun may seemingly increase 
their velocity. The stars beyond the central point will 
exhibit only slow motions, since their great distance from 
our sun will counteract their actually increased velocity. 

In opposition to the great centre the motions would be 
swifter than at our sun. But since only the difference of 
6* 



66 ELEMENTS OF ASTRONOMY. 

their proper motions and the sun's will in most cases be 
known to us, and this only at very considerable actual dis- 
tances, we must only expect feeble motions in opposition. 

More rapid proper motions are to be sought for only at 
great distances from both of these points. How much is 
to be expected in each case will depend upon two circum- 
stances ; the distance of our sun from the central point, 
and the direction of the star's path. 

§ 100. Maedler considers the course of the milky-way, 
according to the determination of other distinguished as- 
tronomers, as pointing out the plane, which is to be taken 
as the ground plane of the starry stratum, and he fixes 
the central point in this plane. This course describes 
nearly a great circle, in the heavens, yet with a devia- 
tion not to be neglected, since the milky-way under the 
meridian of the vernal equinox, at a mean, is 36£° from 
the north pole, while under the meridian of the autumnal 
equinox it is only 26J° from the south pole. This devia- 
tion appears less clearly in the points of intersection of the 
milky-way and the equator, as a consequence of the double 
division (of the milky-way) on one side. It would more- 
over seem that of the two parts into which the milky-way 
divides the celestial sphere, that one is least in which the 
vernal equinox falls, and is farthest from us, our sun being 
on the outside of this plane towards the side of the larger 
part, and we are to seek for the central point toward the 
smaller portion. 

Both the Herschels have demonstrated that we are 
nearer the southern half of the milky-way than the north- 
ern, as is seen by a comparison of the depth of the stars 
in the two directions. Through this circumstance we are 
in a condition to confine our researches for the central 
point to narrower limits, since we must seek it in the north- 
ern part of this smaller portion and between the milky-way 
and the equator. 

We may, however, add that if our sun is not very near 
the central point, (as is far from probable according to 
Herschel's researches on the different depths of stars in the 
milky-way,) on the side of the central point a greater 
crowd of stars is to be expected, although they may not be 



ELEMENTS OF ASTRONOMY. 67 

as dense as those which the region of the milky way pre- 
sents. All regions not thickly strewn with stars we may 
reject. None, however rich and promising, can be admit- 
ted as the true centre unless the proper motions of the 
stars observed and registered point toward it. 

§ 101. From a close examination of the catalogues, 
Maedler has reached the conviction that in the Pleiades 
the central group of the entire starry heavens incluoYed 
within the milky-way is found. He even selects a bright 
star of this constellation, Alcyone, as the individual star 
more likely than any other to be the true central sun. 
Among the compressed groups of the heavens no one is 
found which approaches the Pleiades in brilliancy and rich- 
ness. It is also located in a region very rich in stars, and 
at a point which fulfils very completely the required con- 
ditions. 

Maedler makes however the reservation, that, as in our 
system, notwithstanding the great preponderating mass of 
the central body, the centre of gravity falls without it so 
often as Jupiter and Saturn are separated by less than one 
quadrant from each other ; in like manner the centre of 
gravity of the starry heavens, in the changes among the 
constellations in the lapse of thousands of years, may fall 
without Alcyone, or even pass over to another star. 

§ 102. Assuming Bessel's parallax of the star 61 Cygni 
to be correctly determined, Maedler proceeds to form a 
first approximate estimate of the distance of this central 
body from the solar system. And he arrives at the con- 
clusion that Alcyone is about 34,000,000 times as far re- 
moved from us or from our sun as the latter luminary is 
from us. 

The same approximate determination of distance leads 
to the result that the light of the central sun, if there be 
one, occupies more than five centuries in travelling thence 
to us. The enormous orbit which our own sun is thus in- 
ferred to be describing about the distant eentre, not indeed 
under its influence alone, but by the combined attraction 
of all the stars which are nearer to it than we are, and 
which are estimated to amount to more than 100,000,000 
times the mass of our own solar system, is supposed to re- 



68 ELEMENTS OF ASTRONOMY. 

quire upwards of 18,000,000 years for its complete de- 
scription, at the rate of eight miles for every second of 
time. These calculations are however far from conclusive. 
They do not account for all the proper motions of the stars, 
many of which are in opposite directions, showing rather 
the gravitation round several centres than one universal 
centre. 

§ 103. The motions we have described are not the 
only ones discoverable in the stars. Some of the stars re- 
volve in pairs, or in groups of four or five, round a com- 
mon centre of gravity. These are called binary and mul- 
tiple stars. From the change of brightness in stars appar- 
ently single, motion too small to be otherwise detected is 
inferred. 

Some of these stars entirely disappear, and some vary 
in brightness periodically, while others are irregular in 
their changes. The star Algol passes through a change 
which occupies two days and twenty-two hours. The 
second star in the Lyre goes through its variation in six 
days and nine hours. A star in the Swan, from being vis- 
ible to the naked eye becomes invisible and resumes its 
former brightness in the course of eighteen years. Of the 
stars which are irregular in their changes, some disappear 
suddenly, others undulate in their variations, increase in 
brightness, diminish, and again increase. 

§ 104. Catalogues have not heretofore been sufficiently 
accurate to determine how many stars have reappeared in 
the same places, and how many are to be regarded as new 
stars. It is certain that many formerly recorded in cata- 
logues have disappeared, and that many new comers have 
become permanent dwellers in the heavens, while others 
have shone a while with the utmost brilliancy and then 
disappeared. Observations during the last fifty years, 
show ten or fifteen stars which have either entirely disap- 
peared from their former places, or so completely changed 
their magnitudes as no longer to be registered. Twenty 
have changed in brilliancy, either greatly increased or di- 
minished. Several have become visible of a size and posi- 
tion too striking to have been overlooked if before so bright. 
One of the fifth magnitude, and consequently visible to 



ELEMENTS OF ASTRONOMY. 69 

the naked eye, has made its appearance in the present 
year, 1848. 

Sir John Herschel gives the following account of the 
changes which the star vj Argus has undergone within a 
few years. In a catalogue of the stars made in 1677, it 
is marked as of the fourth magnitude ; in subsequent cata- 
logues, it is put down as of the second. When first ob- 
served by Sir John Herschel, in 1834, it appeared a very 
large star of the second, or a very small star of the first 
magnitude, and thus remained without any apparent in- 
crease or change up to nearly the end of 1837. On the 
16th of December, 1837, Sir John Herschel was struck 
by the appearance of a very bright star in a part of the 
heavens with which he was perfectly familiar. On exami- 
nation it proved to be v\ Argus, shining however with a 
light nearly tripled. From this time until the second of 
January, 1838, it continued to increase. After this it de- 
creased rapidly. Again, in 1843, it increased, and in 
1845, when last particularly observed, was again on the 
decline. 

§ 105. There are now seven or eight authentic records 
of the sudden appearance and subsequent extinction of new 
and brilliant fixed stars. They have once or twice ap- 
peared so suddenly as to strike the eye even of the multi- 
tude. One of the most remarkable instances occurred to 
Tycho Brahe. On Nov. 11th, 1572, as he was walking 
through the fields, he was astonished to observe a new star 
in the constellation Cassiopeia, beaming with a radiance 
quite unwonted in that part of the heavens. Suspecting 
some disease or delusion about his eyes, he went up to a 
group of peasants to ascertain if they saw it, and found 
them gazing at it with as much astonishment as himself. 
He went to his instruments, and fixed its place, from which 
it never afterwards appeared to deviate. For some time 
it increased in brightness — greatly surpassed Sirius in 
lustre, and even Jupiter ; it was seen by good eyes even 
in the day time, a thing which happens only to Venus un- 
der most favorable circumstances, and at night it pierced 
through clouds which obscured the rest of the stars. After 
reaching its greatest brightness, it again diminished, passed 



70 ELEMENTS OF ASTRONOMY. 

through all degrees of visible magnitude, and finally disap- 
peared. Some years after, a phenomenon, equally impos- 
ing took place in another part of the heavens, manifesting 
precisely the same succession of appearances. We are 
quite baffled to account for these astonishing displays. If 
the bodies in question are moving in orbits, how singular, 
that no change of position was observable, and how tre- 
mendous the velocity which could sweep these stars in so 
brief an interval from a region comparatively near to us to 
the invisible depths of heaven. From a comparison of 
records, there is some ground for supposing that the star 
seen by Tycho is not a stranger, but one which appeared 
before, passing through its mighty phases in about 800 
years. If this be so, it ought to appear in about twenty or 
thirty years. 

§ 108. Many of these changes may be accounted for 
by supposing stars to move in orbits more or less elongated. 
Some of the orbits appear nearly circular ; others resemble 
those of comets ; some occupy a few days ; others cannot 
be passed through in less than centuries. If a star were 
receding from us in a straight line we could detect its mo- 
tion only by the diminution of its brightness, and if it re- 
cedes sufficiently far it may become invisible. If it moves 
in an orbit, it will appear again near its former place, and 
perhaps be regarded as a new star. The lost Pleiad may 
thus have receded to a distant part of its orbit, and may 
hereafter reappear. When the period is extremely short, 
and the star disappears, it has been suggested that, revers- 
ing the law of our system, it circles round a non-luminous 
body. When the period of a star occupies some years, we 
have no difficulty in attributing its dimness or disappear- 
ance to its greater remoteness. But when the period of a 
star's variations occupies only three days, it seems proba- 
ble that some other cause of the variation in light exists. 
Perhaps further observations will show that for these va- 
rious phenomena a variety of causes evists. Two other 
causes have been suggested besides those already men- 
tioned. One is the existence of cosmical clouds, which may 
for years lie between a star and us, and dim its light ; 
the other is the occurrence of changes at the surface 



ELEMENTS OF ASTRONOMY. 71 

of the star. Both of these explanations have this recom- 
mendation, that they account for the changes we see in 
the color of the stars as well as those which take place in 
their brilliancy. 

§ 107. The most careless observer must be struck 
with the variety in the hues of the larger stars, and the 
telescope reveals similar differences in all. Some are of a 
deep blue, reel, or yellow, and not the least beautiful are 
of a clear sparkling white. In the southern hemisphere 
there are two planetary nebulae which have of themselves 
one a pale but decided, and the other a more striking and 
intense blue color. Looking back to a great distance of 
time we find the color of some of the stars has changed. 
In the time of Plotemy, Sirius was classed with the five 
other red stars, Arcturus, Aldebaran, Pollux, Antares, and 
a Orionis. It is now of a brilliant white. Clouds of 
some partially opaque substance may have given Sirius 
its reel color, as they may also cause the variations in 
lustre of those stars which do not change periodically. 

§ 108. The phenomena both of motion and color may 
be best studied in the double stars. To these curiosity has 
directed the telescope ever since its first invention. The 
places of many thousands have been ascertained. The 
distribution of the stars was always a question of great in- 
terest. Astronomers sought to know whether they were 
related, whether they revolved round one another, or round 
one great centre. It was early observed that many stars 
were nearer to one another than they would be if they had 
been accidentally scattered in the heavens, thus making it 
probable that some law caused their proximity. The tel- 
escope showed that some of those stars whose brightness 
particularly attracts us, consist of two stars very near one 
another and which appear but one to the naked eye. In 
some cases the proximity is only apparent, they lie one be- 
hind another, nearly in the same visual line to us, but sep- 
arated by millions of miles. But double stars occur in too 
great numbers for us to suppose a mere coincidence of di- 
rection, and the great law of gravity enables us to account 
for their proximity. In most cases two stars, called binary 



72 ELEMENTS 0E ASTRONOMY. 

or double, revolve round their common centre of gravity. 
Sometimes what to the eye appears one consists of three 
or more such stars, in fact is a well-balanced republic, mov- 
ing harmoniously round the common centre. 

§ 109. Computations have been made of the positions 
of the double stars, allowing that they move in an orbit 
governed by gravity. The theoretical and observed posi- 
tions are then compared. Not more than one or two have 
completed their revolutions since they were first observed, 
and there is no sufficient evidence that the same orbit has 
been retraced in successive revolutions. The periods as- 
signed to them are from forty to twelve thousand years. 

In studying the stars which move in orbits, the embar- 
rassment is slight compared with those which either move 
with us onward or backward, or take their apparent motion 
from ours. A sun moving round a dark centre w T ould de- 
scribe an ellipse. If this ellipse were at right-angles to 
the radius of our visual sphere, we should see it in its true 
proportions. If it were inclined differently, it would be 
foreshortened, and the ellipse would change its form. If 
it were in the same plane with the radius, the star w r ould 
appear to vibrate back and forth in a straight line. From 
any of these appearances the actual form of the ellipse 
could be ascertained. 

When both stars are visible, as in the binary systems, it 
is a little more difficult to conceive of the motions. Each 
star moves round the common centre of gravity, and this 
common centre is the focus of each ellipse. The two stars 
being usually of unequal size, describe unequal ellipses, 
but their mutual attraction makes these ellipses similar, and 
causes the bodies to be in similar points of each ellipse at 
the same moment, and consequently to lie opposite to one 
another in a straight line drawn through the centre of 
gravity. Motion has also been detected in the systems of 
three, four, five, and even six stars. 

§ 110. Binary stars are among the most interesting 
objects in the heavens. The two stars are seldom of the 
same color, and rarely of colors at all similar. If one is 
red, the other is usually green ; if one is white, the com- 



ELEMENTS OF ASTKONOMY. 73 

panion is dusky. They are also usually unequal in size. 
The large star is usually orange-red or yellow, and the 
small star blue, purple or green. Imagine the state of the 
planets, if any there be, attached to these suns ! Think of 
being exposed to the alternations of blue and yellow days, 
and to every variation of color arising from the presence of 
one sun, of the other, and of the two together ! 

These colors were at first supposed to be an optical illu- 
sion. A ray of white light contains all the colors of the 
spectrum. When the eye is fatigued by gazing at a color, 
a ray of white light falling on it finds it insensible to that 
color. It excites, therefore, the perception of that color 
only which the eye is capable of perceiving, that is the 
color most unlike the other, its complementary color. The 
apparent color of some of the stars may be owing to this 
fact, but in many cases the color of each star remains 
unchanged when viewed alone. Whatever be this myste- 
rious power by which stars thus connected seem to divide 
the spectrum, there can be no doubt of its existence. 

In a Centauri, the most superb double star in the heav- 
ens, both stars are of a ruddy or orange color, though the 
smaller has a more sombre hue than the larger. The prin- 
cipal star is of the first magnitude, the companion is be- 
tween the first and second. When the light of this double 
star is exceedingly magnified, it is difficult to assign its 
exact size to the second star ; it is apt to appear larger 
than reality. If the light of both be weakened by reflec- 
tion, the difference between them is more evident. The 
diameter of the relative orbit of these stars about each other 
cannot be so small as that of the orbit of Saturn about the 
sun, and exceeds, in all probability, that of the orbit of 
Uranus. 

7 



74 ELEMENTS OF ASTRONOMY. 

CHAPTER VI. 

THE SOLAR SYSTEM. 

The Primary and Secondary Planets. Law of distances from the Sun. 
Revolutions of the Planets in their Orbits, and their revolutions on their 
Axes. Elements of their Orbits. Amount of Solar heat received by each 
Planet. Orbits described by the Secondary Planets. Mass and Densi- 
ties of the Sun and Planets. Eclipses. General views of the Solar 
System. 

§111. Let us return from these distant orbits to the 
only fixed star with whose immediate neighborhood we are 
acquainted. The other stars may or may not be accompa- 
nied with silent and invisible attendants, but we know 
that our sun is surrounded by a train whose minute size 
and dim reflected light must conceal them from distant 
observers whose organ of vision or whose telescopes are not 
greatly superior to our own. 

Eighteen primary and twenty secondary planets com- 
pose the known solar system. It contains no other bodies 
of any considerable mass or they would disclose themselves 
by disturbing the motions of the planets. There are how- 
ever two sets of bodies, meteors and comets, which have 
some claim to be considered members of the system ; and 
which will be described hereafter. 

§ 112. We are now more particularly interested in 
the primary planets, of which the earth is one, and the sec- 
ondary planets or moons which revolve round their prima- 
ries while they accompany them round the sun. Seven of 
the primary and two of the secondary planets have been 
discovered within the last three years, and better instru- 
ments and more extended observations may hereafter reveal 
others. 

Beginning with the planets nearest the sun, they are 
Mercury, Venus, the Earth, Mars, Juno, Ceres, Pallas, 
Yesta, Astraea, Iris, Hebe, Flora, Metis, and Hygeia, 
called from their small size, the Asteroids ; Jupiter, Saturn, 
Uranus, called also Herschel, and Neptune. 



ELEMENTS OF ASTRONOMY. 75 

The Earth is accompanied by one moon ; Jupiter by- 
four ; Saturn by eight, and by two or more rings ; Uranus 
by six and perhaps by more ; Neptune by one, and prob- 
ably more. The planets near the sun are so involved in 
its rays, and those extremely remote are so dim, that 
their satellites, if they have any, are likely to remain 
undiscovered. 

§ 113. A singular proportion exists in the planetary 
distances. The interval between any two is twice as great 
as the inferior interval, and only one half the superior 
interval. The interval between the orbits of Venus and 
the Earth is twice the distance between Venus and Mer- 
cury, and one half of that between the Earth and Mars. 

Let a represent the distance of Mercury from the Sun, 
b the interval between Mercury and Venus, then 

a = 37,000,000=the distance of Mercury. 

a+2°b= 68,000,000 = 

a+2 J b= 95,000,000 = 

a+2*b= 142,000,000= 

a+2 3 b= 288,000,000 = 

a+2±b= 485,000,000= 

a-f-2 5 b= 890,000,000 = 
a+2 6 b=l,843,000,000 = 

But the distance of Neptune, the outermost known planet, 
differs widely from this law. 

§ 1 14. The relative sizes and distances of the sun and 
planets cannot be accurately represented by orreries. We 
may form an idea of their proportions by drawing a circle 
representing the sun's disc, and placing within it all the 
primaries and secondaries of their true proportional size. 

The following calculations give the true proportions of 
the size and distances of the planets. It is better, by 
stretching the conceptive faculty, to imagine these and 
other celestial facts, than by machinery to get a partial or 
false impression. 

No machine can truly represent the motions of the heav- 
enly bodies, because they are acted upon by so many influ- 
ences at once, so that their motions are not what they 
would be in obedience to a single law. Machines are 



u 


Venus. 


a 


Earth. 


a 


Mars. 


a 


Asteroids. 


a 


Jupiter. 


u 


Saturn. 


u 


Uranus. 



76 ELEMENTS OF ASTRONOMY. 

liable to another objection ; they act by contact, by impulse, 
by visible material connection. In the heavens we see no 
such connection or contact between the parts which act on 
one another. The student is advised therefore to imagine 
to himself the relative sizes and distances and motions now 
to be described, in order to strengthen his mind to grasp 
the more complex phenomena to be introduced hereafter. 
An humble part of imagination, the conceptive faculty, 
from want of encouragement is apt to die out before those 
studies are pursued in which it is most needed. Those 
who can bring vividly before the mind the motion and 
mutual actions of the heavenly bodies, find Astronomy easy, 
enjoy and realize its grandeur ; those whose conceptions are 
dull and perplexed, may hear its sublimest truths in vain. 
This faculty, the power of conceiving figures and motions is 
in the power of all. 

§ 115. Let the earth be represented by a globe 1 J 
inches in diameter, the proportionate diameters of the other 
planets would be as follows : — 

Mercury, £ of an inch. 

Yenus, 1 inch. 

Mars, 3 of an inch. 

Jupiter, 16 inches. 

Saturn, 15 " 

Uranus, 6 " 

Neptune, 5 " 

While the Sun would be represented by a massive globe 

whose diameter would be 14 feet. 

Preserving the same scale for the distances of the planets 

from the Sun they would be : — 

Mercury, 190 yards. 

Venus, . . 360 " 

Earth, 500 " 

Mars, 760 " 

Asteroids, 1,400 " 

Jupiter, 2,600 " 

Saturn, 4,800 " 

Uranus, ........ 9,500 " 

Neptune, . . . 15,000 yards, or 8£ miles. 



ELEMENTS OF ASTRONOMY. 77 

Assuming the rate of motion of a common ball to be in 
round numbers 1,000 miles an hour, or about 17 miles a 
minute, to traverse the distance between the Sun and Mer- 
cury would require its uniform flight during about 4 J years ; 
from the Sun to Venus, 8 years ; to the Earth, nearly 11 
years ; to Mars, 16 years ; to the Asteroids, 28 years ; to 
Jupiter, 59 years ; to Saturn, 102 years ; to Uranus, 200 
years ; and to Neptune, 300 years. 

§ 116. Since the Sun is 330,000 times as massive as 
the Earth, bodies are drawn to him at equal distances with 
a force 330,000 times as great as terrestrial gravitation. 
Yet so distant from him are the planets that if let fall from 
their mean distances towards the Sun they would occupy 
in falling to it the following times : — 

Days. Hours. 

Mercury, 15 13 

Venus, 39 17 

The Earth, 64 13 

Mars, 121 10 

Vesta 235 



Juno, 281 5 

Ceres, 297 6 

Pallas, 301 4 

Jupiter,. ....... 765 19 

Saturn, 1,901 

Uranus, 5,425 

Neptune, 10,600 

The Moon would fall to the Earth in 4 20 

§ 117. The Sun, and all the planets yet accurately 
examined, have spheroidal forms, differing from one another 
only in being more or less flattened at the poles. They 
have a motion of rotation round a fixed axis, from west to 
east, which is also common to some at least of the satel- 
lites. The planets and satellites have likewise a motion of 
revolution round the sun in the same direction as their rota- 
tion from west to east. The axis of revolution, like that of 
rotation, is always parallel to itself. In different planets it 
points toward different quarters of the heavens. The paths 
described by the planets in their revolutions are ellipses, 
differing only in being more or less elongated. The paths 
7* 



78 ELEMENTS OF ASTRONOMY. 

described by the satellites are ellipses in respect to their 
primaries, but as at the same time they move round the 
sun also, their real path is compounded of these two 
motions. 

The planets nearest the sun revolve more rapidly than 
those more distant. They also rotate more slowly. Thus 
while their year is shorter, their day is longer, than that of 
more distant planets. Rotation has not been ascertained 
of the two outer planets, though it may be presumed. The 
following table shows the periods of rotation and of revolu- 
tion which have been ascertained. 



Mercury, 

Venus, 

Earth, 

Mars, 

Asteroids, 

Jupiter, 

Saturn, 

Urunus, 

Neptune, 

§ 118. Not only is the rotation of the more distant 
orbs more rapid, but the orbs themselves are larger. 
Their equatorial particles therefore describe very large 
circles with extreme rapidity, moving much faster than the 
equatorial particles of the earth. The rate at which a 
man at the equator of the earth rotates is 8,000x3= 
24,000 miles divided by 24,=1,000 miles an hour. An 
inhabitant of Jupiter rotates 90,000x3=270,000 miles 
in 10 of our hours, or 27,000 miles an hour. 

The day and year of the planets have here been reckon- 
ed in days and years of the earth. Since the day of the 
outer planets is much shorter than ours, its year contains 
many more of its own clays than it would of ours. Jupi- 
ter's year contains 10,000 of his own days. Saturn's years 
30,000 of his days. 

§ 119. The ellipses described by the planets approach 
very nearly to circles. The excentricities in parts of the 
semi-axes are : — 



Day and Night. 


Time of revolution. 


Hours, Minutes. 


Years. 


Months 


. Days. 


24 6 





2 


28 


23 21 





7 


15 


23 56 


1 





2 


24 39 


1 


10 


21 


unknown 


4 





21 


9 56 


11 


10 


17 


10 29 


29 


5 


24 


9 30 


84 





37 


unknown 


164 








ELEMENTS OF ASTRONOMY. 79 

Mercury, 0.2055149 Ceres, 0.0784390 

Venus, 0.0068607 Pallas, 0.2416480 

Earth, 0.0167836 Jupiter, 0.0481621 

Mars, 0.0933070 Saturn, 0.0561505 

Vesta, 0.0891300 Uranus, 0.0466794 

Juno, 0.2578480 Neptune, 0.0087195 

§ 1 20. The velocities in miles per second with which 
the planets move in their orbits are as follows : — 

Mercury, 30 miles. Ceres, 11 miles. 

Venus, 23 " Pallas, 11 " 

Earth, 19 " Jupiter, 8 " 

Mars, 15 " Saturn, 6 " 

Vesta, 13 " Uranus, 4 " 

Juno, 12 " Neptune, 3 « 

§ 121. We have seen that the orbits of the planets 
differ as to size and form, they also are performed in dif- 
ferent planes. These planes do not vary much from the 
plane of the sun's equator. If we imagine the plane of the 
sun's equator to be extended throughout the solar system, 
the planets and moons will in one part of their orbits be on 
one side of this plane, in the other part on the other side 
of it. 

The axis of rotation in all planets which have been 
closely observed, does not coincide with the axis of revolu- 
tion ; consequently the plane of each planet's rotation 
differs from that of its revolution. 

The plane of the sun's rotation also is inclined to its 
orbit. The motion of revolution of the sun is a motion 
forced upon it by the planets. As they move round the 
common centre of gravity the sun cannot remain stationary. 
By virtue of his greater mass he remains near the centre 
of the system, and compels planets and moons to circle 
round him. But in return, their united imfluence forces 
on him a slight irregular motion round a point which 
always lies near his own surface. 

This revolution is performed in an imaginary plane, 
called the fixed ecliptic, determinable from the velocities 
and the masses of the planets, which like the centre of 
inertia, never changes its position, on account of the mutual 



80 ELEMENTS OF ASTRONOMY. 

actions of the bodies of the system. This plane of inertia 
is called the fixed ecliptic. Its situation is nearly half way 
between the orbits of Jupiter and Saturn. It is inclined 
at a small angle only to the plane of the earth's orbit, 
which is called the earth's ecliptic. 

§ 122. It is more convenient to compare the planes 
of the planets with the plane of the earth's orbit or the 
ecliptic, than with the plane of the sun's equator. The 
following table gives the inclination of each planet's orbit 
to the ecliptic, and the inclination of the planet's equator 
to its orbit : — 

Inclination of planet's 
Inclination of orbit to ecliptic. equator to its orbit. 

Mercury, 7° " 

Venus, 3° 24' 75° 

Earth, 0° 00' 23° 28' 

Mars, 1° 51' 29° 30' 
Asteroids, 7° 8', 13° 5', 10° 37', 34° 37', 

Jupiter, 1° 19' 3° 5' 

Saturn, 2° 30' 27° 

Uranus, 0° 46' " 

Neptune, 1° 47' " 

Sun, 7° 10' 

Moon, 5° 8' 0° 

The satellites of Jupiter are inclined to Jupiter's orbit : 

The First, 3° 5' 30" 

The Fourth, 2° 58' 48" 

The orbits of the seven interior satellites of Saturn are 
nearly circular, and very nearly in the plane of the ring. 
That of the eighth is considerably inclined to the rest, 
and approaches nearer to coincidence with the ecliptic. 

The orbits of the six satellites of Uranus are inclined 
about 78° 58' to the ecliptic, and their motion is retro- 
grade. The orbits appear to be nearly circles. 

§ 123. The rapid succession of day and night in the 
remote planets, may, by the activity which it excites, mod- 
ify the torpidity caused by the length of the year and by 
the great distance from the sun. At Mercury the sun 
shines with seven times the intensity experienced on earth, 



ELEMENTS OF ASTRONOMY. 81 

and at Uranus his radiation is at least three hundred and 
thirty times weaker than with us. Between Mercury and 
Uranus there is an actual disproportion in the quantity of 
solar light shed upon them of upwards of two thousand to 
one. Yet Uranus is not obscure ; it receives as much 
light at noon-day as if nearly one thousand of our pale 
moons were shining in its sky. Neptune, in a given space, 
receives about tudtt P ar ^ °f the light which the earth 
receives. 

§ 124. Some idea may be formed of the effect its 
greater distance has upon the climate of each planet, by 
considering the sun's apparent size as viewed from each 
planet. 



Mercury, would be 


85' 


Venus, " 


46' 


Earth, " 


32' 


Mars, " 


21' 


Vesta, " 


13' 


Juno, " 


12' 


Ceres, " 


11' 


Pallas, " 


11' 


Jupiter, " 


6' 


Saturn, " 


3' 


Uranus, " 


1' 


Neptune, " 


1' 



§ 125. How far the atmospheres of the nearest and 
most remote planets may modify the otherwise intense heat 
and cold we know not, nor what affect they may have on 
organic life. Creations different from those we know may 
people these globes, and no more perceivable by our means 
and senses than the animal is by the vegetable world. We 
do not even know the effect of our atmosphere on ourselves. 
We know not what baneful or what blessed influences it 
excludes. We are born within it, we can never lay it 
aside or judge how far it modifies our perception of all 
beyond it. But we may safely infer from analogy that the 
sun is, in those worlds, as in our own, the source of light, 
heat, growth and motion. Through his influence the winds 
blow, the waters inundate, the earth is clothed with verdure 
and prepared for the habitation of man. 



82 ELEMENTS OF ASTRONOMY. 

§ 126. In these remote regions we find also, as some 
compensation, more of those satellites which so much adorn 
them. Mercury, Venus and Mars know only the stars ; 
but Jupiter has four moons, each larger than ours, con- 
stantly circling around him and varying his skies. Saturn 
has eight and Uranus six. Neptune is also attended by 
one and probably by more. 

The moons of Jupiter revolve in Id. 18hs. ; 3d. 13hs. ; 
7d. 4hs. ; 16d. ifhs. 

Those of Saturn in 2d. 3hs. ; Id. 9hs. ; Id. 21hs. ; 2d. 
18hs. ; 4d. 12hs. ; 15d. 23hs. ; 21d. ; 80cl. 

Those of Uranus in 5d. 21hs. ; 8d. IT'hs. ; lOd. 23hs. ; 
13d. llhs. ; 38d. 2hs. ; 107d. 17hs. 

All but three of these periods are shorter than our lunar 
month, and most of the orbits are very much larger, so 
that the moons display immense activity, and a rapidly 
changing series of phases and eclipses. 

Unless they had rapid motions of their own, giving them 
energetic tendencies to fly off, the immense attraction of 
the vast globes round which they revolve, would absorb 
them in their mass. 

§ 127. Although the satellites are usually spoken of as 
revolving around their primaries, this is not strictly the 
truth. Each planet with its satellite perpetually keeps 
itself balanced on each side of the common centre of grav- 
ity, and it is this centre of gravity, which, properly speak- 
ing, moves round the sun. Thus the moon forces the 
earth to adjust itself at such a distance from the centre 
of gravity as to balance itself. Thus the path of Jupiter 
and of each of his moons undergoes continual modifications 
in order to preserve the centre of gravity. If the centre 
of gravity is preserved, and one of the bodies, the satellite, 
has a revolving motion, the earth also must slightly revolve 
or sway nearer and then farther from the sun than the 
centre of gravity. Since this is the case the moon must 
be each half month, alternately, nearer to and then farther 
from the sun than the earth ; the earth therefore is each 
half month farther from and then nearer to the sun than 
the centre of gravity is. 

The periods of rotation of the satellites as far as ascer- 



ELEMENTS OF ASTRONOMY. 83 

tained are equal to the the times of their revolution. Con- 
sequently these bodies always turn the same face to their 
primaries. 

§ 128. The manner in which the equal times of 
the moon's rotations and revolutions bring the same face 
always present to the earth, may be seen by moving 
round a centre without rotating it, a ball painted half white 
and half black. If its white face is turned toward the sun 
when in one position, and it is then moved onward without 
rotating, when it has performed one quarter of its revolu- 
tion, only a half of the white face will be toward the centre. 
When it has performed half the circuit none of the white 
face will be toward the centre. Thus without rotation the 
white face cannot remain visible from the centre. But if 
the ball roll slowly round keeping presented to the centre 
as large a proportion of its equator as it passes through of 
its orbit in a given time, it will finish its rotation and 
revolution in the same time and have the same face always 
toward the centre. 

In the same way a person who begins to ascend a cir- 
cular flight of stairs with his back toward a certain wall, 
finds himself obliged to rotate once in the course of his 
ascent on reaching the top his back is toward the same 
wall, but it has been in every other direction during the 
ascent. If he tries not to rotate but keeps his back obsti- 
nately in the same direction, his face cannot always be 
toward the centre. The rotations and revolutions are 
always in the same direction from west to east with the 
exception of the satellites of Uranus. 

§ 129. As we know but little of the more distant sat- 
ellites, and as the phenomena appear the same in all, with 
the exception above mentioned, we will now confine our- 
selves to the earth's moon. 

The plane of the moon's revolution is inclined to the 
ecliptic 5° 8'. It moves eastward at the rate of two thou- 
sand miles an hour and completes its revolution in twenty 
nine days. 

The axis of the moon's rotation is inclined to the pole of 
its orbit, and always preserves its parallelism with itself. 

Beside the moon's rotation, and revolution round the 



84 ELEMENTS OF ASTRONOMY. 

earth, it revolves round the sun, as the earth does, in a 
large ellipse. If it felt only the earth's attraction, it 
would describe an ellipse with the earth in one of the foci, 
returning to the same place at the end of each month. 
But as it performs at the same time a small ellipse round 
the earth and a large ellipse round the sun, the path really 
described in space is a compound of these two motions, it 
is a succesion of curves, varying in concavity, but always 
concave toward the sun. 

§ 130. Let us imagine two persons fastened together 
by a rod of a certain length which compels them to keep 
always at the same distance, one from another. Let them 
both describe a large circle round a tree, and let the 
smaller one at the same time go round his companion. 
And let the companion walk so rapidly, that the other can 
never actually return to his former place but is forced to 
move on. 

We have here a rude image of the moon's motion round 
the sun. The earth, rushing a million and a half miles 
daily in her orbit, bears on the moon and straightens out 
the curve she would otherwise describe. In some parts of 
the moon's monthly orbit her course is accelerated, in 
others it is delayed, and the curve consequently varies in 
concavity. 

Let us begin with the moon between the sun and earth, 
the new moon. During the first quarter of her orbit, the 
moon's revolution round the earth will tend to carry her 
backward in space. Her revolution round the sun tends 
to carry her onward. The former motion will tend to 
delay the latter. In the next two quarters of her monthly 
revolution, the two motions will coincide in direction, and 
the result will be greater rapidity of motion. Fig. 2d, 
Plate I., gives the motion of the moon for one month. 

Jupiter is so large and so near to his satellites, in com- 
parison with the sun, that the curves which they describe 
are different from the path described by our moon, although 
they go round Jupiter as the moon goes round the earth. 

§ 131. Let A, B, C, D, E, Fig. 3, Plate I, be as 
much of Jupiter's orbit as he describes in eighteen days ; 
and the curves a, b, c, d, will be the paths of his four moons 



ELEMENTS OF ASTRONOMY. 85 

going round him in his progressive motion. The first satel- 
lite intersects its own path once in 42 J hours, making such 
loops as those in the diagram. The second satellite, mov- 
ing more slowly, crosses its own path once in three days, 
thirteen hours, making out five loops in the time in which 
the first makes ten. The third satellite, moving still more 
slowly, comes to an angle at the end of seven days, four 
hours, and then describes another such curve. The fourth 
satellite is always progressive, making neither loops nor 
angles in the heavens. 

Those satellites whose velocities round their primaries 
are greater than the velocities of their primaries in open 
space, make loops when nearest to the sun. This is the 
case with Jupiter's first and second satellites, and with 
Saturn's first. 

But those satellites whose velocities are less than the 
velocities of their primary planets, move direct in their 
whole circumvolutions. This is the case with the third 
and fourth satellites of Jupiter, and with the second, third, 
fourth and fifth, satellites of Saturn, as well as with our 
moon. 

As the moon turns upon her axis in precisely the same 
time which she takes to revolve about the earth, she has 
but one day or night in one of our lunar months ; and as 
she encompasses the earth thirteen times during the earth's 
progress round the sun, it is manifest that a lunar year 
contains about thirteen lunar clays. 

§ 132. There are striking differences in the relative 
sizes and weights of the planets. And the different pro- 
portions of their size and weight cause essential differences 
in their material composition. 

The following table shows the known sizes and densities 
of the members of the solar system. 

Mercury. Venus. Earth. Mars. Jupiter. Saturn. Uranus. Sun. Moon. 

Diameter in 
miles, 3140 7700 7916 4100 90000 76000 35000 883000 2160 

Volume, that of 
the earth be- 
ing one, 0.06 0.93 1.00 0.14 1470 887 77 132S460 0.02 

Mass, that of 
the earth be- 
ing one, 0.16 0.94 1 0.13 33S 120 17 354936 0.013 

Density, that of 
the earth be- 
ing one, 2.95 .99 1 0.79 0.25 0.11 0.26 9.26 .75 



86 ELEMENTS OF ASTRONOMY. 

In comparing the size of bodies, we must observe whether 
their diameters, their discs, or their solid contents are com- 
pared. The discs are to each other as the squares, and 
the volumes as the cubes of the diameters. 

The mass of the sun bears to the mass of the earth but 
a small ratio compared with that which its volume bears to 
the volume of the earth. In judging the volume of the 
sun, we take the extent of the bright surface, which proba- 
bly is an atmosphere. This atmosphere may be many 
thousands of miles deep, and of course has a low specific 
gravity. The density ascribed to the sun is however com- 
posed of the density of this atmosphere and of that of the 
sun's body, which may be very great. The sun proper, 
without its atmosphere, as we calculate the planets, would 
have a smaller size and a greater density. 

§ 133. The densities of two bodies are directly as 
their masses or weights, and inversely as their volumes. 
More frequently the density is compared with that of a 
globe of water of the same size, and the specific gravity is 
thus obtained. Thus the Sun weighs 1J compared with a 
globe of water of the same size ; Mercury 17 T V ; Venus 
5| ; Earth 4 T 9 ^ ; Mars 3 T 3 o ; Jupiter 1 T V ; Saturn J ; 
Uranus 1. 

The matter of which Mercury is made is nearly four 
times as heavy as that which composes our earth, while 
Saturn is as light as cork or deal. Mercury has the den- 
sity of quick-silver ; Uranus and the Earth, that of steel ; 
Mars and the Moon, that of diamond ; the Sun and Jupi- 
ter, nearly that of resin. Upon the weight or mass of 
each planet depends, chiefly, the weight of bodies near its 
surface. 

§ 134. If we suppose two planets, of equal masses, 
but one of one hundred times the density of the other, a 
man at the surface of the smaller one would weigh most, 
because he would be nearest the centre of gravity. If 
these two planets were rotating in the same time, and the 
man stood near the equator, the difference of weight would 
still be increased, because the surface which performed the 
largest circle would generate the most centrifugal force, 
and thus diminish his weight most. Weight on the surface 



ELEMENTS OE ASTRONOMY. 87 

of a planet at rest is in direct proportion to the mass of the 
planet, and in inverse proportion to the square of the dis- 
tance from the centre. On two bodies of unequal density, 
but equal size, weight is greatest at the densest ; on two of 
equal density, but unequal size, greatest at the largest. 
Weight on a large, not dense body, may be just equal to 
weight on a small dense one. If both bodies rotate, the 
weight at the surface of each decreases rapidly, in propor- 
tion to the rapidity of rotation. Thus the larger planets 
exercise less attraction on bodies at their equators, on ac- 
count of great rapidity of rotation. 

§ 135. The number of particles in the attracting mass 
being changed, the conditions not only of inorganic but of 
organic creations are altered. There is a limit to the size 
of animals, trees, buildings, in each planet. No house can 
stand when made so large or so loosely that the cohesive 
force of its parts does not overcome the attractive force of 
the earth. No animals now exist on earth so large as 
those which reposed in the marshes of the primitive world. 
The sea yet has its whales, because it bears up their 
bodies, and thus as it were diminishes gravitation. If man, 
with his present organization, were transported to the sur- 
face of a body as large as the Sun, he would probably fall 
to pieces like a figure of smoke. If the cohesion of his 
body were increased, he would yet be unable to move, un- 
less greater muscular power were granted. For the at- 
tractive force of the Sun would cause bodies to fall through 
334 feet in a second, and would consequently attract man 
thirty times as strongly, and give him thirty times as much 
weight as he has here. A man of moderate size would 
weigh about two tons at the surface of the sun. Whereas 
at one of the asteroids, he would weigh but a few pounds, 
and would find it difficult to remain attached to the planet. 

§ 136. We have considered the Sun as the controller 
of the system and its motions, and as the dispenser of light 
and heat ; we will now consider him as he clothes other 
bodies with light. Not only all the day-light, but all the 
planet and moon-light of our system comes from him. As 
the train revolve, both of primaries and secondaries, one 
half of each orb is lighted up by him. One half of each is 



88 ELEMENTS OF ASTRONOMY. 

always in light, one half in shade. When two planets, or 
a moon and a planet are in such a position that the light 
of the Sun is reflected from one to the other, the former 
becomes visible by reflected light. Whenever a planet or 
a moon passes between another planet and the Sun, the 
Sun's light will be cut off, at least partially, from the 
second planet. We will consider what interferences can 
arise in the solar system, and as we view them from the 
earth we will mention those visible from the earth, and 
give them the names usually applied. 

A body may disappear on account of another body's 
coming between it and the source of its light, or on account 
of another body's coming between it and the spectator. In 
the former case, the body is really eclipsed or darkened ; 
in the latter case, it remains illumined, but is no longer 
seen by us. We apply the term eclipse to both of these 
occurrences, though they differ widely in their nature. 
A real eclipse is the same viewed from all parts of the sys- 
tem ; an apparent eclipse is only an eclipse to one particu- 
lar place, sometimes tmlv to u«^ jjuttkm of the smkh, aU a 
time. 

§ 137. Eclipses of the first kind take place when the 
moon passes into the earth's shadow ; or when any of the 
satellites enter the shadow of their primaries. 

If the planets were so large, or so near to one another, 
that the shadow cast away from the sun by an inner one 
could reach the surface of an outer one, the outer one 
would be eclipsed whenever they passed in their orbits ; 
but their small size and mutual distance forbids this. 

Partial eclipses of this kind take place whenever the 
satellites pass between their primaries and the sun, and 
cast their shadows on the discs of their primaries. On 
such occasions, to those portions of the planet on which the 
shadow falls, the sun appears eclipsed. 

Eclipses of the second kind occur : — 

When the moon passes between the sun and the earth, 
and the sun is eclipsed to the earth. When the moon 
passes between a fixed star or a planet and the earth, cuts 
off its light and occults it. Or what is less observable, 
when the sun occults a planet or a star. Planets some- 
times, but rarely, eclipse one another. 



ELEMENTS OP ASTRONOMY. 89 

When Mercury or Venus passes between the earth and 
the sun, and intercepts from our view a small portion of his 
disc, it then appears on the sun's disc as a little black ball, 
and its passage is called a transit. 

In like manner the satellites of Jupiter may conceal 
from us a portion of Jupiter's illuminated surface, or Jupi- 
ter may conceal from us its satellite's illuminated surface. 
A satellite of Saturn may pass between its ring and the 
earth, or Saturn or his ring may occult a satellite. 

Eclipses of the first kind, take place when the sun, the 
interposing, and the eclipsed body are in one straight line. 
Eclipses of the second kind take place when the observer, 
the interposing, and the eclipsed body are in one straight 
line. 

§ 138. If the various members of the solar system 
moved in the same plane eclipses would take place much 
more frequently than they now do. Mars and Venus 
would pass between the earth and the sun almost once in 
each one of their revolutions. The moon would eclipse the 
sun every month, and would itself pass into the earth's 
shadow every month. As all the planes are inclined to 
one another it is but rarely that the centres of any three of 
the bodies are in one straight line. 

Bodies in the heavens are often invisible to an observer on 
earth, from two other causes, because they are lost in the 
superior light of a neighboring body, and because they are 
so situated that their bright surface can not send any rays 
to the earth. When Mercury and Venus are nearly in a 
straight line between the sun and earth, the greater part of 
their surface which is toward the centre, is dark and there- 
fore invisible ; the only part visible is a portion of the illu- 
minated surface. When the planets are nearly in a line 
with the earth and sun, but beyond the sun, their bright 
surface is toward us but is lost in his rays. 
- When the moon is between the earth and the sun, her 
bright side is toward the sun and she is invisible for some 
hours. When the moon is beyond the earth her whole illu- 
minated hemisphere is visible from the earth. In all situa- 
tions between these two, a portion of her illuminated hemis- 
phere is visible from the earth. 
8* 



90 ELEMENTS OE ASTRONOMY* 

Of course only opaque bodies, such as shine with reflect- 
ed light can become invisible from this cause. The moon's 
dark side might perhaps be visible here as it is said to be 
in Syria, and as we see the old moon in the new moon's 
arms, if it were not when new so near the sun. 

§ 139. We have now brought before us one by one all 
the circumstances of the solar system. We have imagined 
the sun balancing by his mass the revolving planets. These 
with their satellites, each in its particular plane, and with 
its own velocity, never resting, moving from the beginning 
of its creation, and turning on its axis, all obediently 
circle round the sun. We have seen each with its pole of 
rotation invariably pointed toward the same star, wherever 
the motion of revolution bears it on. One half of each orb 
is bathed in light, the other plunged in darkness. All are 
always in the starlight, but the stars are seen by none till 
it has turned into its own shadow. 

For three thousand millions of miles on every side the 
obedient orbs recognize the central power. Beyond this 
may lie other subjects, but their reflected light is too dim 
to attract our eyes. We may hereafter learn their exis- 
tance from the perturbations of Uranus and Neptune. 

We have found that the members of the solar system dif- 
fer in many important particulars ; in distance from the 
sun ; in times of rotation, and of revolution ; in mass ; in 
density ; in degree of compression at the poles ; in the 
planes and ellipticities of their orbits ; and in the inclination 
of the plane of the rotation to that of the revolution of 
•each planet. Most of these circumstances must affect 
greatly the physical condition of the surface of each planet. 
Many minor causes also, as the nature of the atmosphere 
or its absence, the presence of water and many unknown 
causes doubtless introduce still greater variety. 

§ 140. Some of these differences observed among the 
planets appear to follow a law, and doubtless a law always 
exists though undiscernible by us. The planets within the 
Asteroids are of more moderate size, are more dense, 
rotate round their axes more slowly, and in nearly equal pe- 
riods, and are less compressed at the pole than the planets 
beyond the Asteroids, and with one exception, are without 



ELEMENTS OF ASTRONOMY. 91 

satellites. The outer planets are of much greater magni- 
tude, are less dense, more than twice as rapid in their rota- 
tion, more compressed at their poles, and possess all but 
one of the satellites of the system. These remarks cannot 
however be applied strictly to each planet. Nor are 
there any constant relations between the distances of the 
planets from the sun, and their absolute magnitudes, den- 
sities, times of rotation, eccentricities, and inclinations of 
orbit and of axis. Neither in size nor density is there any 
regular succession as we go from the sun. The time of 
rotation decreases on the whole with the increasing solar 
distance, yet it is greater in Mars than in the earth, and in 
Saturn than in Jupiter. Juno, Pallas and Mercury have 
the greatest eccentricity ; and Venus and the earth which 
come between them have the least. Nor is there more 
regularity in the inclination of the orbits, or the position of 
their axes of rotation relatively to their orbits ; though on 
the whole those planets which have the most elongated 
orbits, have their orbits most inclined to the ecliptic. The 
orbits of the different planets are elongated in different 
directions ; the position of the major axis of each orbit is 
not however invariable. 

§ 141. In the relative size of the moons and primaries 
no law is discoverable. The earth's moon is of great relative 
magnitude, its' diameter being to that of the earth, as one 
to four, whereas the diameter of the largest of all known 
satellites, the sixth of Saturn, is but one seventeenth, and 
and that of Jupiter's largest satellite is but one twenty-sixth 
part of the respective diameters of the planets round which 
they revolve. 

The density of the moon is three fourths that of the earth, 
while the second satellite of Jupiter appears to be actually 
more dense than the great planet round which it revolves. 

The satellites of Saturn offer the greatest contrasts both 
of absolute magnitude, and of distance from the central 
planet. The sixth satellite is but little smaller than Mars, 
(whose diameter is twice that of our moon) while the 
recently discovered satellite is one of the smallest bodies in 
the solar system. 

The absolute distance of a satellite from its primary is 



92 ELEMENTS OP ASTRONOMY. 

greatest in case of the outermost satellite of Saturn. It is 
above two millions of geographical miles, or ten times the 
distance of our moon from the earth. The satellite which 
is nearest to its planet is undoubtedly the innermost of 
Saturn, and it offers the only example of a period of revolu- 
tion of less than twenty four hours. Its distance from the 
centre of the planet is eighty thousand and eighty eight 
miles ; from the surface of the planet it is forty seven thou- 
sand four hundred and eighty miles. 

§ 142. If we estimate distances not in absolute meas- 
ure but in radii of the primary planets, we find that the 
nearest of Jupiter's satellites, which in absolute distance is 
twenty six thousand miles farther from the centre of that 
planet than our moon is from the earth, is only six radii of 
Jupiter from its centre, while our moon is distant from us 
fully sixty and a half radii of the earth. 

Even the law, mentioned above, of the distances of the 
planets from the sun is not numerically exact for the dis- 
tances between Mercury, Yenus and the earth, and is violat- 
ed in the case of Neptune. Even allowing this law to have 
no exceptions it is one found only by observation ; we 
have no idea on what principle it is founded, nor how it 
acts. 

But there are circumstances in the form and motions of 
the planets whose principle and immediate cause are known 
to us. These are of the deepest interest because they 
throw light on the past condition of the planets. The 
spheroidal form of the sun and planets, their two motions 
in the same direction, the near coincidence of the planes of 
their revolution with the plane of the sun's equator, give 
us a hint as to what forces presided at their birth. 



ELEMENTS OF ASTRONOMY. 93 

CHAPTER VII. 

METEORS AND THE ZODIACAL LIGHT. - 

Appearance and number of Meteors ; their composition and size. Mete- 
oric showers ; their supposed origin. The Zodiacal Light ; its appear- 
ance ; different theories of its nature; its possible connection with Mete- 
oric showers. 

§ 143, Thus far we have dealt with facts. All we 
have learned has been from observation and reasoning. If 
we would go farther and inquire what circumstances exert- 
ed the forces we have been tracing, we enter on the domain 
of theory. A theory which would explain the formation of 
the system must however include all its members, and 
there are some members of our solar system which we 
have not yet described. Let us make ourselves acquaint- 
ed Trffch tlveaa 5 swaxl tko» wo -glxsill La prepared in include 

in one view, all the forms which matter, to our eyes, ever 
assumes. 

There are two more bodies or classes of bodies which 
decidedly belong to our system ; the Zodiacal Light and 
Shooting Stars or Meteors ; the latter have even been 
claimed as belonging to the earth's atmosphere. Beside 
these are ether, which perhaps is common to our system 
and to the rest of space, of which we know nothing except 
that it is unlike every other form of matter, and comets, 
which we can scarcely claim as belonging exclusively to 
our system, but which exhibit matter under very peculiar 
and interesting conditions. 

Having studied these objects and also the physical state 
of the nearer planets, we shall have the slight data on 
which all theories of the formation and former state of the 
universe are founded. 

§ 144. Shooting Stars have a particular interest : 
they are the only visitants from other worlds which ever 
reach the surface of our earth ; the only foreign matter 
which enters the earth's atmosphere and may be touched 



94 ELEMENTS OF ASTRONOMY. 

and examined. They give us therefore our only intelli- 
gence of the physical composition of the rest of the solar 
system. 

We become conscious of the presence of most members 
of the solar system either by the light they send us, or by 
the attraction they exercise on us as we pass them in 
space. We know the presence of comets because they 
send us light, but we do not know it from their attrac- 
tion, because if they exercise any on us it is so slight as to 
be imperceptible. 

Now it is possible that, in the immense fields of space 
through which we journey, there may be other travellers, 
whose minute size prevents our detecting them either by 
sight or through gravity. When we pass near these small 
bodies we may have no intimation of their presence. But 
if they enter our atmosphere, drawn by the earth's attrac- 
tion, they may become luminous, and either be dissipated 
in the upper regions of the air, or fall to the ground. 
They would thus present to us the phenomena of meteors. 

^ 1 4/5- Tin o largo motooro arc* called, glebes UI" flry- 

balls. The small meteors exhibit only a bright path 
or line, and are called Shooting Stars. These balls and 
stars often appear together. In general they have the same 
hue as the fixed stars. Their color sometimes becomes 
yellow, and sometimes blue or green. The trains which 
they leave behind them are not smoky, but rather like a 
shower of sparks. Sometimes the star breaks into frag- 
ments which form a continuation of the train, and which 
vanish almost as soon as that. When they break they 
sometimes let fall stony fragments, covered with a distinct 
shining black crust, such as our ovens could not produce. 
Though considerably heated they are not incandescent. 
They sometimes appear to have been softened, but never 
to have been melted during their passage through the air. 
Wherever they have been collected, in all periods of time, 
and in all parts of the earth, they resemble one another in 
their form, in the nature of their crust and in their chem- 
ical composition. About one third of the elementary sub- 
stances which compose our globe have been found in them. 



ELEMENTS OF ASTRONOMY. 95 

They are generally composed of metals, among which 
nickel, cobalt and virgin iron are the most common, or 
they are clayey and contain crystals. They fall with a 
force which causes them to sink from ten to fifteen feet 
into the earth. Their form proves that they are frag- 
ments. None have been known to fall more than seven or 
seven and a half feet in diameter. They are usually of 
much smaller size, and many seem never to reach the 
earth, and either to have no perceptible mass, or to be en- 
tirely dissipated in the atmosphere. These may perhaps 
have small nuclei surrounded by inflammable vapors or 
gases. Some of the fire balls which appear the largest 
may be of this kind. The apparent size and brightness 
seem to have no connexion with the size of the fragments 
let fall. 

§ 146. Meteors are visible in great numbers and in 
all parts of the heavens. A register kept from July 1841 
to February 1845 gives five thousand three hundred and 
two Shooting Stars, observed in one thousand and fifty 
four hours, from one observatory. Among these were 
eight globes, and eighty Shooting Stars of the first magni- 
tude. Whence it follows that an observer would see one 
globe a week, and one falling star of the first magnitude, 
each night of eleven hours. 

Single meteors such as we have described appear in all 
quarters of the heavens. They vary as to swiftness and 
as to height ; some being not more than sixteen and others 
one hundred and forty miles high. The largest appear to 
have the greatest altitude, and only the smaller ones ap- 
pear to come within twenty or even forty miles of the 
earth. The motion in all cases is not in the same direc- 
tion ; the prevailing direction is from north east to south 
west, contrary to the motion of the earth in its orbit. 
This direction of the motion is particularly observed in 
those meteors which fall in showers. They come from the 
same point during the whole continuance of a shower, 
which proves their independence of the earth's rotation, 
and consequently that they come from without our atmos- 
phere. This fact and the periodical recurrence of the 
showers has given to meteors a new importance. It has 



96 ELEMENTS OF ASTRONOMY. 

made improbable the before received theories concerning 
their origin ; unless indeed we suppose that there may be 
several kinds produced by different causes. It is not well 
ascertained of what importance these periodical showers 
are, but they are too striking and peculiar to be over- 
looked, and they may throw light on other unexplained 
phenomena. 

§ 147. Single meteors or even showers of stars ap- 
pearing irregularly have been explained by supposing them 
to be gaseous substances condensed in the upper regions of 
the atmosphere, perhaps by the same agency which con- 
densed the earth. Their composition, of metals found also 
in the earth, favored this hypothesis. 

The hypothesis of their lunar origin has also found some 
believers. The great size and height and the crater-like 
form of the lunar mountains led to the supposition that 
they were extremely active. A body projected from a vol- 
cano in the moon, with a velocity of about eight thousand 
five hundred feet a second, would not fall back on the 
lunar surface, but would recede from it indefinitely. In 
order to reach the earth it would require a velocity of only 
eight thousand three hundred feet. Such a velocity, which 
is only about four or five times that of a cannon ball, is 
quite conceivable. But the extraordinary exhibitions of 
1799 and 1833 are quite irreconcilable with a lunar 
origin. 

To be satisfactory, a theory must explain not only their 
coming in showers, but the periodical recurrence of the 
showers in the months of August and November. 

It is only within seventy years that attention has been 
directed to this subject and though on looking back some 
traces of periodicity have been found, they are scarcely 
sufficient to establish their periodicity as a law. The prin- 
cipal displays have been in 1799, 1832, 1833 and 1844. 

On the 11th of November, 1799, thousands of shooting 
stars were observed by Humboldt, at Cumana ; and on the 
same night by different persons, over the whole continent 
of America, from Brazil to Labrador, and also in Germany. 
In 1832 they were seen over the whole of the north of 
Europe, and in 1833 the wonderful display took place in 



ELEMENTS OF ASTRONOMY. 97 

North America, which has been so well described by Pro- 
fessor Olmsted, and which first established the importance 
of the subject. He thus describes the great meteoric 
shower of the 13th of November, of that year. 

§ 148. " On that morning, from 2 o'clock until broad 
daylight, the sky being perfectly serene and cloudless, the 
whole heavens were lighted with a magnificent display of 
celestial fire works. At times, the air was filled with 
streaks of light occasioned by fiery particles darting down 
so swiftly as to leave the impression of their light on the 
eye, (like a match ignited and whirled before the face,) 
and drifting to the north west like flakes of snow driven by 
the wind ; while, at short intervals, balls of fire, varying in 
size from minute points to bodies larger than Jupiter and 
Yenus, and in a few instances, as large as the full moon, 
descended more slowly along the arch of the sky, often 
leaving after them long trains of light, which were, in some 
instances, variegated with different prismatic colors. 

On tracing back the lines of direction in which the me- 
teors moved, it was found that they all appeared to radiate 
from the same point, which was situated near one of the 
stars, (Gramma Leonis) of the Sickle, in the constellation 
Leo ; and in every repetition of the meteoric shower of 
November, the radiant point has occupied nearly the same 
situation. 

This shower pervaded nearly the whole of North Amer- 
ica, having appeared in almost equal splendor, from the 
British Possessions on the north, to the West India Islands 
and Mexico on the south, and from sixty one degrees of 
longitude east of the American coast, quite to the Pacific 
Ocean on the west. Throughout this immense region, the 
direction was nearly the same. The meteors began to at- 
tract attention by their unusual frequency and brilliancy, 
from nine to twelve, in the evening ; were most striking in 
their appearance from two to four; arrived at their maxi- 
mum, in many places, about four o'clock; and continued 
until rendered invisible by the light of day. The meteors 
moved in right lines, or in such apparent curves, as, upon 
optical principles, can be resolved into right lines. Their 
general tendency was toward the north-west, although, by 
• 9 



98 ELEMENTS OF ASTRONOMY. 

the effect of perspective, they appeared to move in all 
directions. 

§ 149. It is considered as established that the me- 
teors had their origin beyond the limits of the atmosphere, 
having descended to us from some body existing in space 
independent of the earth ; that they consisted of exceed- 
ingly light combustible matter ; that they moved with very 
great velocities, amounting in some instances to not less 
than fourteen miles per second ; that some of them were 
bodies of large size, probably several hundred feet in 
diameter ; that when they entered the atmosphere, they 
rapidly and powerfully condensed the air before them, and 
thus elicited the heat which set them on fire, as a spark is 
sometimes evolved by condensing air suddenly by a piston 
and cylinder ; and that they were consumed and dissolved 
into small clouds at the height of about thirty miles above 
the earth." 

Professor Olmsted referred this periodical return to 
astronomical causes and predicted its return at the same 
season, in future years. It was visible in different parts of 
the earth every year until 1839, and since then it has 
ceased altogether. 

§ 1,50. The following is Professor Olmsted's reason- 
ing, and his theory. 

Since the earth fell in with the meteoric body in the 
same part of its orbit, several years in succession, the body 
must either have remained there during a year, or it must 
itself have had a revolution round the sun. No body can 
remain stationary in the planetary spaces, or it would be 
drawn either into some nearer body or into the sun. The 
body whence meteors fall must therefore have revolved 
either in a year or some aliquot part of a year, or it could 
not have come in contact with the earth so many successive 
years. If it revolves in an elliptic Orbit it will some years 
encounter the earth and other years pass at a distance 
from it. This may explain the absence of the showers for 
several years. The meteoric body is too small to be seen. 
It probably consists of myriads of planetoids, which, for all 
we know, may fill the planetary space. They may circu- 
late about the sun, generally in groups or zones, and two 



ELEMENTS OF ASTRONOMY. 99 

of the zones may intersect that part of the earth's orbit 
through which it passes in August and November. When 
the earth encounters a thin portion of the zone the showers 
are scanty and if the intervals in the zone are wide, only 
scattered meteors will be visible, as on ordinary nights. 

This zone of planetoids may be as old as the larger 
planets, and may rank as an important portion of the sys- 
tem. It may consist of those portions of matter which 
were not sufficiently near one another to be attracted into 
one mass. 

It is possible that this revolving zone may be composed 
not of solid bodies, but of nebulous matter like the tails of 
comets. We can more easily understand the disappear- 
ance of nebulous than of solid matter in our atmosphere, 
and a very large proportion of meteors never touch the 
earth. 

It has also been suggested that meteors may have their 
origin in the zodiacal light, a phenomenon hereafter to be 
described. Since the plane of this nebulous substance is 
not parallel to the ecliptic, the earth might pass through it 
at one season, and be remote from it another. But this 
does not account for the appearance of shooting stars at all 
seasons of the year. 

The interruption of these phenomena may be caused 
by a motion of the nodes of the stream of aerolites, so that 
what has at former periods been so striking, and what has 
been repeated in our own times, will again recur after an 
interval. 

§ 151. The zodiacal light and meteors, although very 
unlike one another in appearance, may perhaps arise from 
similar causes. The zodiacal light is a pale cone of light 
projected from the sun after the evening and before the 
morning twilight. It is almost constantly visible in the 
torrid zone, but in northern temperate regions, is only 
distinctly visible in the beginning of spring, after the eve- 
ning twilight, and at the end of autumn before the com- 
mencement of the morning twilight. Its light resembles 
that of a comet. The faintest stars may be seen through 
it. It is less bright than the milky way, with ill denned 
edges, scarcely to be distinguished from twilight. Hum- 



100 ELEMENTS OF ASTRONOMY. 

boldt describes it in 10° latitude as appearing very regu- 
larly about an hour after sunset. Before this, even if the 
night was perfectly dark, no trace of it could be seen. 
Then it suddenly became visible, extending in great bright- 
ness and beauty between Aldebaran and the Pleiades and 
attaining an altitude of 39°. Long narrow clouds appear- 
ed low down on the horizon, as if in front of a golden cur- 
tain, while bright tints played on the upper clouds. The 
light diffused in that part of the heavens appeared almost 
to equal that of the moon in her first quarter. When its 
brightness was greatest a mild reflected glow was visible in 
the east. Towards ten o'clock it became very faint, and 
at midnight only a trace of it remained. 

§ 152. Its figure agrees with that of a spheroid seen 
in profile. It has the sun for its base, and its axis lies 
nearly in the direction of the zodiac whence it takes its 
name. It also lies very nearly in the plane of the sun's 
equator. 

As the sun's equator is differently inclined to the hori- 
zon, on account of the different positions of the sun in 
the ecliptic, the zodiacal light inclines with it, and is in a 
great measure concealed beneath the horizon ; or at least 
its lustre is diminished by vapors. In the vernal equinox 
the arc of the ecliptic which the sun is about to enter is 
more elevated above the horizon of a place in north latitude 
than the equator is. The zodiacal light is then elevated 
above the equator by all the obliquity of the ecliptic ; no 
other position is so favorable for our climate. In the summer 
solstice the arc of the ecliptic, and consequently the lumi- 
nous cone, is parallel to the equator, and therefore much 
more inclined to the horizon of places \r\ north latitude thar* 
in the spring. 

The apparent angular distance of its vertex from the sun 
varies according to circumstances, from 40° to 90° ; and 
the breadth of its base perpendicular to its axis varies from 
8° to 30°. It must involve Mercury and sometimes Venus 
and the Earth, and if it were not extremely rare, would 
produce some disturbance in their motion, but in fact it 
does not appear to impede the progress even of the tails of 
comets, 



ELEMENTS OF ASTRONOMY. 101 

As to its probable composition we must choose between 
the supposition of its being purely nebulous, or loaded with 
the tails of millions of comets ; or of its consisting of a 
stream of countless planetoids or meteors, too small to be 
seen separately, but able from their numbers to give out 
a faint light. This latter hypothesis has this advantage, 
that it resembles the cause assigned for periodic shooting 
stars. 



CHAPTER VIII. 

COMETS. 

The number of recorded Cornels. Variety in their motions and appearance. 
Their immense size. Description of a Comet. The tails of Comets. 
Bessel's Theory of their formation. Halley's Comet. Biela's and Encke's. 
Their resistance by the ether. The mutual influence of Comets and 
Planets. Mass of the Comet of 1770. The probable effect of a collision 
•with a Comet. 

§ 153. Comets form a class of bodies entirely distinct 
from the fixed stars and from the planets, whether we 
regard the character of their movements or their physical 
constitution. They receive their name from their hairy 
appearance, caused by the coma or atmosphere which 
surrounds them. Of the number of comets it is impossible 
to speak with certainty. Many comets on their nearest 
approach to the sun are too distant to be seen from 
the earth ; many may not have reached their perihelion 
within the recorded experience of man \ many may be in- 
visible from their diminutive size ; many can be seen only 
from the south side of the equator, where there are but 
few means of observation ; many, though on the north side 
of the equator, rise above the horizon only during the day ; 
many pass unnoticed, owing to cloudy weather. Several 
have however been seen so bright as to be visible in the 
day time, even at noon and in bright sunshine ; and there 
9* 



102 ELEMENTS OE ASTRONOMY* 

is oiie instance on record of a very large one observed 
near the stinj when eclipsed, in the year 60 before Christ. 
The number of comets which enter our system must amount 
to many thousands; more than six hundred have been 
actually observed, and the orbits of between one and two 
hundred have been calculated. 

They come from every region of the heavens, and move 
in every variety of plane. Some move in the same direc- 
tion with the planets ; others in the opposite direction. 
Some of them remain in sight for a few days only, others 
for many months ; some move with extreme slowness, others 
with extraordinary velocity. Not unfrequently the two 
extremes of apparent speed are exhibited by the same 
body, in different parts of its course. 

§ 154. Not only does a comet vary in its physical 
appearance and its speed in different parts of one course, 
but it sometimes presents on its return, an appearance so 
different as to be scarcely recognizable. Its size and 
splendor are sometimes so much diminished that it is diffi- 
cult to identify it. 

We must not however suppose that all the apparent 
changes in the tails of comets are real. Many of them are 
owing to the state of our atmosphere, as is proved by the 
same comet's appearing of different brilliancy and extent, in 
different parts of the globe. Our atmosphere is a coarse 
medium, through which to view objects so delicate. * 

Another circumstance which makes it difficult to identify 
a comet is, that their orbits are liable to change after they 
enter the solar system, owing to the attraction of the plan- 
ets. The orbit of a comet is, however, more to be relied 
on, as a test of its identity, than its physical appearance, 
if the changes in its orbit can be accounted for by the in- 
fluence of any member of the planetary system. 

§ 155. We will now consider, in detail, the physical 
constitution of comets, their motions and the influence they 
may impart to, or receive from, the members of the solar 
system. 

Comets are the most voluminous and at the same time 
the lightest bodies of our system. 

The tails of some of the largest have extended over a 



ELEMENTS OE ASTRONOMY. 103 

distance of from thirty to forty million leagues, a length 
much exceeding the interval between the sun and the earth. 
At the same time their weight is so slight that not one has 
disturbed a planet or a satellite, in the slightest perceptible 
degree. It follows that matter so exceedingly diffused 
must be transparent. Stars of the sixteenth and seven- 
teenth magnitude, which the slightest fog would conceal, 
may be seen through their substance ; and yet their light 
passes through thousands of miles of the body of the 
comet. 

It has been doubted whether comets shine with a light 
of their own or by reflected light like planets. As they 
present no phases like the moon, it has been supposed that 
they originate light. They may however reflect it through- 
out their whole substance like the light clouds of our at- 
mosphere, which often appear soaked in light. The fact 
that part of their light is polarized makes it certain that at 
least part of their light is reflected. 

§ 156. Comets consist of a large ill defined mass of 
cloudy luminous matter, usually increasing in brilliancy 
toward the centre. This central portion is called the 
nucleus of the body. The nucleus and coma belong to 
comets in all parts of their orbit. The tail and head are 
developed as they approach the sun. 

The Nucleus of the comet is generally to be distinguish- 
ed by its forming a comparatively bright point in the centre 
of the head. In most instances it has the appearance 
of a solid body, and frequently subtends an angle capable 
of telescopic measurement. It is usually enveloped in a 
dense nebulous stratum, called the coma. This stratum so 
frequently renders the edge of the nucleus indistinct, that 
it is extremely difficult to ascertain its diameter with any 
precision. 

But though comets, in general, possess this nucleus or 
body, there are many of them in which it seems to be en- 
tirely wanting, and which present only a nebulous mass, 
having a gradual condensation towards the centre. Appa- 
rently, there is a regular gradation of Comets, from such 
as are composed merely of a gaseous or vapoury medium, 
to those which, by the mutual attraction and consolidation 



104 ELEMENTS OF ASTRONOMY. 

of their nebulous particles, have at length acquired a con- 
sistent nucleus. In the small comet of 1804, for example, 
no solid body could be discovered ; it seemed to consist 
entirely of vapours. A star of the sixth magnitude could 
be distinguished through the very centre of the comet of 
1796 ; and Herschel asserts a similar fact with regard to 
that of 1795. Through the comet of 1802 a star of the 
tenth magnitude, could be observed, with hardly any di- 
minution of its light. The second comet which appeared 
in 1798, was estimated to have a nucleus of twenty-seven 
miles in diameter. The nucleus of the comet seen in De- 
cember,"1805, was computed to be thirty miles in diam- 
eter. The comet of 1799 had a nucleus three hundred 
and seventy three miles in diameter. The first comet of 
1811 had one four hundred and twenty-eight miles in 
diameter ; and the second comet of that year was observed 
to possess a nucleus of prodigious size ; from Herschel's ob- 
servations it was no less than two thousand six hundred 
and thirty-seven miles in diameter, or one third of the 
earth's diameter. 

§ 157. In all comets there is an envelope of light, 
which in some cases seems to be united with the nucleus. 
This envelope almost never surrounds the nucleus, but 
forms a sort of hemispherical cap on the side next the sun, 
and then diverges into two brilliant streams on the opposite 
side, giving rise to the singular and well known phenome- 
non of the comet's tail. Some comets ape furnished with 
several of these singular appendages. That of 1744 had 
six, which spread out like an enormous fan, extending to a 
distance of nearly 30°. 

Many of the brightest comets, however, have been ob- 
served with short and faint tails, and not a few have been 
entirely without them, and in these cases the whole nucleus 
presents only a globular mass of nebulosity. This shining 
envelope is supposed to be of the same nature as the stra- 
tum immediately contiguous to the nucleus, viz : matter 
raised from the surface by the action of the sun's heat, and 
converted into a state of high attenuation. It is found to 
vary considerably, as well in its own thickness as in its 
distance from the nucleus. In the comet of 1811, for ex- 



ELEMENTS OF ASTRONOMY. 105 

ample, the depth of the envelope at one time amounted to 
twenty-five thousand miles, and its distance from the cen- 
tre of the nucleus was found to be thirty thousand miles. 
In the comet of 1807, the depth was thirty thousand miles. 
The small comet of 1804, which seemed to have no solid 
part at all, presented a mass of nebulosity of about five 
thousand miles in diameter. 

Herschel supposes that a very elastic and transparent 
medium surrounds comets like an atmosphere, in which, 
w T hen the cometic matter has become sufficiently rarefied 
by the solar heat, it rises to a certain elevation, and re- 
mains there suspended. The transparency of this atmos- 
phere is proved by the appearance of small stars through 
it. Its elasticity may be inferred from the circular form it 
always assumes. 

§ 158. That the atmosphere of comets must be of 
very considerable extent, is evident both from the great 
depth of the envelope, and from the not unfrequent occur- 
rence of several of these nebulous envelopes one above 
another, all of which must necessarily be suspended in the 
same buoyant medium. As any matter suspended in such 
a medium must have a density inversely proportional to 
its height, it w r oukl follow that the outermost of these 
envelopes should not be so bright as those nearer to the 
nucleus ; and this inference is fully verified by obser- 
vation. 

The extraordinary size of the atmospheres is owing to 
the slightness of the attraction of the exceedingly small 
central mass. If the earth, retaining its present size, were 
reduced, by any internal change, to one thousandth part 
|ts present mass, the atmosphere would expand to more 
than a thousand times its present bulk. 

These nebulous envelopes are thus formed. When the 
comet is approaching the sun the nebulous matter suspend? 
ed in its atmosphere is made to rise higher by the increase 
|ng energy of the sun's heat. For the same reason, after 
one envelope has risen to a considerable height, so much 
matter may subsequently be detached from the nucleus as. 
to constitute a second, which, from being more dense than 
the first, will occupy a lower situation in the atmosphere . 



106 ELEMENTS OF ASTRONOMY. 

In like manner a third and fourth envelope may be success- 
ively formed. 

Thus the comet of 1744, which at its perihelion ap- 
proached the sun to within one fifth the distance of the 
earth, had, about three weeks previous to its perihelion 
passage, a double envelope, and on the seventh or eighth 
day after the passage had acquired a third. As the great 
comet of 1811 receded from the sun, its envelope, losing 
its high degree of attenuation, at length subsided altogether 
upon the nucleus. 

As the tails are much more striking after the comet has 
passed the sun, it has been suggested that the nebulous 
matter when exceedingly excited, may be invisible as 
steam is when it first issues, and afterward being precipi- 
tated, it becomes visible. 

§ 159. The tail is only a continuation of the nebulous 
envelope, which, after nearly encompassing the hemisphere 
of the nucleus next the sun, diverges to a greater or less 
extent in an opposite direction. The tail is sometimes 
wanting, and sometimes is forty or even one hundred mil- 
lions of miles in length. The tail is always of a conical 
shape, the apex being the hemispherical envelope, and the 
base generally ten or twelve times as broad as the diameter 
of the nucleus. 

As the whole envelope is equally exposed to the action 
of the sun, (which in some way or other produces the tail,) 
all the parts are caused by impulsion to assume the shape 
of a conoid, and thus the tail is hollow. Therefore the 
sides or edges of the tail have usually the appearance of 
two brilliant streams, the space between them appearing to 
contain a much less quantity of nebulous matter. For as 
the line of vision traverses a greater number of luminous 
particles at the sides, where that line is a tangent to the 
cone, than toward the middle, where the line of vision is 
more perpendicular to the envelope, there is a much 
greater quantity of light at the sides than at any other 
point. For the same reason, the top of the hemispherical 
cap, or that part nearest the sun, is generally more brilliant 
than any other point. 

Another fact proves at once the hollo wness and the con- 



ELEMENTS OF ASTRONOMY. 107 

ical form of the tail. In whatever position comets are 
placed, and they are frequently observed during a course 
of 180° round the sun, they constantly present the same 
appearance, as to the shape of the tail, and the superior 
brilliancy of its edges. 

§ 160. The length of the tails of comets has some- 
times been enormous. That of 1618 is said to have been 
104° in length ; that of 1680, immediately after its perihe- 
lion passage, was twenty million leagues, and occupied only 
two days in its emission from the comet's body ; a decisive 
proof of its being driven forward by some force, the 
origin of which, to judge from the direction of the tail, must 
be sought in the sun itself. The diameter of the head of 
the comet of 1843 exceeded one hundred thousand miles. 
The breadth of the tail in some places was eight hundred 
thousand, while the extent could not be less than one hun- 
dred and seventy million miles, or nearly equal to the 
diameter of the earth's orbit. 

It is hardly possible that matter once projected to dis- 
tances so enormous should ever be collected again by the 
feeble attraction of such a body as a comet. 

Biela's comet, on its return in 1846, exhibited the aston- 
ishing phenomenon of a double comet. It may have been 
originally double, or may have become so since its last ap- 
pearance, when it was seen undivided, the two portions 
subsequently journeying along side by side, in orbits 
slightly differing from one another, and apparently quite 
undisturbed by any mutual attraction. 

§161. When the comet approaches the sun the nu- 
cleus or densest part never appears in the middle point as it 
certainly would do in obedience to gravity, if the interior 
arrangements of the particles were undisturbed by any 
force from without. Sometimes a vast mass of matter 
streams from the comet in a direction away from the sun; 
sometimes, as in Encke's comet, the body takes an oval 
shape with the nucleus near one edge. As the tail uni- 
formly turns from the sun we cannot doubt that the sun is 
the seat of the disturbing force. 

Does it then drive away the tail or attract the nucleus. 
Evident! v it does neither only. If without compensation the 



108 ELEMENTS OF ASTRONOMY. 

sun repelled any portion of the mass of a comet, the grav- 
itation of the whole comet toward the sun would be dimin- 
ished. If the sun exercised any new or peculiar attraction 
on the nucleus, the gravitation of the whole mass would be 
increased. Either of these effects alone would produce a 
marked change in the comet's motion in its orbit. In the 
former case this would decrease, in the latter it would in- 
crease. Neither of these changes take place. 

§ 162. It occurred to Bessel that the sun might have 
both an attractive and a repulsive influence on the comet, 
one exactly balancing the other ; and although deranging 
the comet's internal constitution, not affecting its gravitat- 
ing tendency or its motions as a whole. 

Many forces act in this way, producing not a single 
effect, but two opposite and compensating ones. Magnet- 
ism is of this kind. If the sun places the comet in a con- 
dition like that in which a magnet places a needle, its grav- 
ity would be undisturbed, but its mass would be endowed 
with polarity. The comet of Halley, during its last return, 
confirmed this view. As it approached the sun, very ex- 
traordinary activities appeared to affect its entire organiza- 
tion. At a very early period there was a singular out- 
streaming of light from the upper part of the nucleus 
towards the sun. This outstreaming mass soon showed 
itself to have a movement of oscillation or vibration exactly 
like that of a pendulum, causing it to swing from one side 
to the other of the line joining the nucleus with our lumi- 
nary. This oscillation could be produced only by an 
attraction exercised by the sun ; just as the swinging of 
the pendulum is owing to the attraction of the earth. It 
must either be a new power, or gravity attracting an irreg- 
ular mass. The times of the oscillation being calculated, 
it was found that it could not be attributed to gravity. It 
is more probable that the sun cast the entire mass into a 
state of polarity. Much excitement was visible, and it 
seemed like a body being magnetized by induction. Bes- 
sel thought that when saturated, the luminous matter 
which had been thrown out toward the sun, turned round 
and enveloping the nucleus formed the tail. 

§ 163. The influence of the sun being doubtless inva- 



ELEMENTS OP ASTRONOMY, 109 

riable, the various effects produced by it on the shape of 
comets, points out varieties in their physical constitution. 
On one he bestows a slightly elongated form, on another a 
tail streaming through spaces wide as the earth's orbit ; 
he causes a third to spread itself out like a fan. 

The following account, given by Sir John Herschel, of 
Halley's comet, as he observed it at the Cape in 1835, 
gives a lively idea of the changes caused in its physical 
appearance by the sun. When first seen by him it ap- 
peared as a star of the third magnitude, hazy, and with: 
a scarcely perceptible tail. The next night a crescent- 
shaped cap was formed on the side next the sun, the coma 
decidedly extending beyond it. These appearances con- 
tinued, with some variations, till it passed its perihelion. 
Twenty-four days after its disappearance it was again seen,, 
in its return from the sun, as a small star of the third 
magnitude, dim and hazy and with no tail. Viewed through 
the twenty-feet reflector, it was now a most surprising ob- 
ject. Its head was terminated sharply like the ground- 
glass shade of an argand lamp. Within the well-defined 
head, and somewhat eccentrically placed, was an object 
resembling a miniature comet, having a nucleus, head, and 
tail of its own, perfectly distinct, and considerably exceed- 
ing in intensity of light the nebulous head. At this time 
the comet was increasing in dimensions with such rapidity 
that it might almost be said to be seen to grow. Measure- 
ments of the diameter of the head taken within two hours 
and a quarter of each other differed sensibly. The next 
night its increase was evident at the first glance ; its form 
had become elongated and less definite toward the tail ; 
the coma had not increased proportionally and was much 
less bright. The whole bulk of the comet, exclusive of the 
coma, had considerably more than doubled within twenty-four 
hours. 

§ 164. By the 31st of January the coma had entirely 
disappeared ; the interior comet was so much dilated as 
more than to fill the field of view (15°) in length, and 
nearly so in breadth ; its outline was soft, rounded, well 
defined. From this night may be dated the commence- 
ment of the developement of the true tail, that is of the 
10 



110 ELEMENTS OF ASTRONOMY. 

prolongation into a regular train of the parabolic envelope, 
aided by the similar prolongation of the ray or internal 
comet. The coma from this time appeared no more ; it 
was neither dissipated nor absorbed, but swept off by the 
sun's action into the tail. But in the progress of the comet 
towards extinction, the semblance of a new coma arose 
from the dilatation of the mass of internal light imme- 
diately surrounding the nucleus, which at last constituted 
the whole visible comet, the infinitely minute and hardly 
perceptible nucleus excepted. 

The comet on the 31st was full of small stars ; but their 
light was not extinguished by it. Innumerable small stars 
passed at various times extremely near to the nucleus, 
though none exactly on it, but were no more affected than 
they would have been by so much lamplight artificially in- 
troduced. 

While the comet was measurable after its perihelion pas- 
sage its dilatations were nearly uniform. Calculating 
backward, at the same rate, the envelope must, on the 21st 
of January, at 10 minutes P. M., have had no magnitude. 
Previous to that instant the comet must have consisted of 
a mere nucleus or stellar point more or less bright, and a 
coma more or less dense and extensive. At that instant 
the formation of this envelope and of the ray or internal 
comet commenced. The perihelion passage took place on 
the 15th of November, and it was not until the eighty-third 
day after that event that the formation of the envelope 
commenced. During these eighty-three days the comet 
must have been cooling, and must have arrived at the 
dew-point of the vaporous substance which composes the 
envelope. 

§ 165. We have now to consider the motions of 
comets. All the planets and all the satellites, so far as we 
know, revolve in orbits of one kind, in ellipses. Among 
comets there is probably a variety. Two small comets re- 
volve in ellipses, and return regularly, and are considered 
as belonging to our system. Probably many others revolve 
in extremely elongated ellipses. Parabolas, hyperbolas, 
and exceedingly elongated ellipses, are so nearly alike at 
the part nearest the sun, that it is difficult to ascertain in 



ELEMENTS OP ASTRONOMY. Ill 

which of these curves a comet moves. Successive obser- 
vations, proving the reappearance of the same comet, may 
prove the elliptic form to prevail. Only two have been 
completely computed whose orbit is best represented by a 
hyperbola. The chance that comets describe parabolas is 
infinitely small compared to that of their describing ellipses 
or hyperbolas. To produce the former curve one particu- 
lar velocity is necessary, the slightest increase or diminu- 
tion of which will cause it to deviate into one or other of 
the two latter curves. 

It may be doubted whether the comets of our system 
have always belonged to it, or whether they were visitants, 
and are detained by some causes unobserved by us. We 
may have an opportunity of settling whether this ever takes 
place by observation at some future time. At present we 
know nothing of their history, we know only that they now 
revolve about the sun in regular ellipses. 

§ 166. The periodic time of Halley's comet is between 
seventy-five and seventy-six years. Its orbit is a length- 
ened ellipse, extending far beyond that of the planet Ura- 
nus, and inclined to the ecliptic at an angle of between 
17° and 18°. Its aphelion is about the distance of Venus 
from the sun. The next is the comet of Encke ; its revo- 
lution is completed in 1,207 days, or 3-i- years. It revolves 
in an ellipse of great eccentricity, at an angle of about 13° 
22' to the plane of the ecliptic. The third has a period of 
six years and eight months. The orbit of Biela's comet 
extends somewhat beyond Jupiter. At its perihelion, how- 
ever, it approaches nearer to the sun than the earth. 
Encke's comet, at its perihelion, is about as distant from 
the sun as Mercury ; at its aphelion, not quite so far as 
Jupiter. The number of known periodical comets is yearly 
increasing. Halley's comet, in its first recorded appear- 
ances, in 1305, 1456, &c, exhibited a brilliant tail. The 
apparent size and the length of the tail seem to have un- 
dergone diminution in its later returns. Encke's comet is 
not at all conspicuous, and Biela's is also small and without 
a tail, or any appearance of a solid nucleus whatever. 
The orbit of Biela very nearly intersects that of the earth, 
and had the latter, at the time of its passage in 1832, 



112 ELEMENTS OF ASTRONOMY. 

been a month in advance of its actual place, it would have 
passed through the comet. 

§ 167. These periodical comets are of particular in- 
terest, because they have almost and will hereafter quite 
settle a question much agitated among philosophers ; 
whether the inter-planetary space is filled with an extremely 
subtle medium, or whether it is a vacuum. Some phe- 
nomena make either probable. If we receive the undula- 
tory theory of light, as is now almost universally done, we 
require a medium stretching not only through the planets 
but to the remotest stars ; for how otherwise can the un- 
dulations be propogated. On the other hand, if such a 
medium exists, we should suppose that the planets would 
have their motions altered by it in a degree which would 
in process of time become appreciable. But no period is 
so constant as the revolution of a planet. Each one ac- 
complishes its revolution, and has done so for hundreds of 
years, in precisely the same time. A medium, however, 
which would impress no delay on the solid, weighty mass 
of a planet, may produce a very perceptible difference in 
the time of the revolution of a comet. Now it was found 
by Encke that the comet which bears his name had been 
constantly anticipating the calculated time of its arrival at 
its perihelion ; in some instances two days, in others one 
day. Its ellipses are continually diminishing, its mean dis- 
tance from the sun dwindling by slow but regular degrees. 
This is evidently the effect which would be produced by 
the resistance of a very rare ethereal medium pervading 
the regions through which comets move. For such resist- 
ance, by diminishing its actual velocity, would diminish its 
centrifugal force, and thus allow the sun power to draw it 
nearer. There is no other mode of accounting for the 
phenomenon in question, and accordingly it is the solution 
generally received. The comet will probably ultimately 
fall into the sun, should it not first be dissipated altogether, 
a calculation not at all improbable, considering the light- 
ness of its materials. 

Of comets not yet returned, we cannot know whether to 
consider them of our system or not. Perhaps they pass 
the long void which separates one fixed star from another. 



ELEMENTS OF ASTRONOMY. 113 

returning after ages to the same centre. Perhaps if the 
sun is translated in space, comets formerly visible may be 
left out of the sphere of his attraction, and the sun enter- 
ing new groups or streams of comets may give them new 
orbits. It has even been supposed, by Arago, that a great 
number of comets might at a distance assume a nebulous 
appearance. What is the density of this ether and the 
law of its density near the sun, whether it is at rest or in 
motion, if the latter, in what direction it moves, are ques- 
tions which comets must answer for us. If it revolves 
round the sun it must accelerate some comets and retard 
others ; and by repeated observations on comets moving in 
different planes and diameters, the plane and rate of rota- 
tion may be ascertained. Halley's comet has been retarded 
in every successive return, and this explanation of its delay 
has been offered ; but we know but little about it. 

§ 168. The periods of those comets which reappear 
more seldom can be determined with great difficulty, and 
as yet with no exactness. A period of 3,065 years has 
been assigned to the fine comet of 1811, and one of up- 
wards of 8,000 years to the comet of 1680. If these pe- 
riods are the true ones, these bodies recede to distances 
from the sun equal, one to twenty-one, and the other to 
forty-four times the distance of Uranus, or to 33,600 and 
70,400 millions of miles. Even at these distances they 
feel the sun's attraction. But while the motion of the 
comet of 1680 at its perihelion is 212 miles a second, 
thirteen times that of the earth, its velocity at its aphelion 
is scarcely ten feet in a second, only three times that of the 
most sluggish rivers, and one half that of the Cassiguian, 
an arm of the Orinoco. It approaches 163 times as near 
to the sun as the earth does, and experiences a heat 
26,000 times that of the earth ; and since red-hot iron has 
only twelve times the heat received from the summer tropi- 
cal sun, this comet is exposed to a heat 2,000 times as 
great as red-hot iron. This comet's distance from the sun 
is only forty-four times that of Uranus, while the nearest 
fixed star has 250 times that distance. Outside of this 
perhaps many comets revolve whose major axes are longer 
than that of the comet of 1680. 
10* 



114 ELEMENTS OP ASTRONOMY. 

Having considered the greatest known distances of 
comets, we will now notice instances of their greatest prox- 
imity hitherto measured. This same comet of 1680 ap- 
proached the sun's surface within one sixth of his diameter. 
Perihelia which take place beyond the orbit of Mars cannot 
often be observed from the earth, yet we have no reason to 
suppose that more lie within than without it. We have an 
opportunity therefore to observe very few of those comets 
which actually enter the solar system. 

§ 169. The mutual influence of comets and planets 
has always been a subject of great interest. By these the 
comet's progress may be retarded or accelerated, the place 
of its nodes changed, its perihelion distance diminished or 
increased, and the inclination as well as eccentricity of its 
orbit altered. And these changes during one revolution 
are sometimes so considerable as to render the identity of 
& comet at its successive returns to the sun very doubtful. 

Halley's comet first drew the attention of astronomers to 
these perturbations. After having ascertained its ap- 
proaches to the sun in the years 1531, 1607, and 1682, 
Halley was surprised to find that the period of its first revo- 
lution was longer by thirteen months than the following 
one. He thought the difference might be caused by the 
disturbing action of the planets, particularly Jupiter and 
Saturn ; and after a rough estimate of their attractions he 
announced the return of the comet for the end of 1758 or 
the beginning of 1759. The comet appeared as announced, 
proving the weight of comets and the extent of gravitation 
to them. During its next revolution this comet will be 
very much diverted from its course by the planet Uranus. 

§ 170. The comet of 1770 exhibited still more re- 
markable changes in its orbit. Astronomers had in vain 
endeavored to represent its observed course by a parabola. 
At length its orbit was discovered to be an ellipse, not so 
elongated as to approximate to a parabola, but much 
shorter, and requiring only a period of five and a half 
years. This result seemed very extraordinary, since the 
comet which should to have been so often visible, on account 
of the shortness of its period and perihelion distance, had 
never yet been seen on any previous occasion ; and the 



ELEMENTS OF ASTRONOMY. 115 

circumstance was still more unaccountable, when it was 
found that the comet made no subsequent return to the 
sun. 

At length, by tracing back the movements of this comet 
in its orbit, it was found that at the beginning of 1767 it 
had entered within the sphere of Jupiter's attraction. The 
amount of this attraction being calculated from the known 
proximity of the two bodies, the previous orbit of the comet 
was determined. It must have been an ellipse of greater 
extent, having a period of fifty years, in which the comet, 
even when nearest the sun, was still as far distant as Jupi- 
ter. It was therefore very evident, why, as long as the 
comet continued to circulate in this orbit so far from the 
centre of the system, it never became visible from the 
earth ; and also that the cause of its appearance in 1770 
was the disturbing action of Jupiter which constrained it 
to move in a shorter ellipse and at a less distance from the 
sun. 

§ 171. Another question of interest is whether comets 
can act on planets so as to produce perturbations in their 
course ; and also what would be the consequence of a col- 
lision between a comet and a planet. The comet of 1770 
affords an answer to the first question. From its brilliancy 
this comet must have been of considerable size, and was 
even computed to have a diameter nearly thirteen times as 
large as the moon. On the two occasions above mentioned 
it is said to have traversed the whole system of Jupiter's 
satellites, and at each time required four months to free 
itself from the sphere of his attraction. Yet not the 
slightest alteration was observed in the motions of these 
small bodies. The same comet approached so near the 
earth as to shorten its own revolution by two days, yet 
what was its reaction on the earth ? If its mass had been 
equal to that of the earth, it would have lengthened our 
year by two hours, forty-seven minutes. But nice calcu- 
lations prove that in the length of that year no change ex- 
ceeding two seconds would have taken place. Hence as 
10,027" : 2" : : mass of earth : mass of comet. The 
comet's mass was less than the -^Vu part of the mass of the 
earth. It is evident therefore that none of the planets are 



116 ELEMENTS OF ASTRONOMY. 

liable to be carried out of their course by these diminutive 
bodies. 

Other dangers have been apprehended from the ap- 
proach of comets. It has been feared that the waters of 
the ocean would be attracted and thus form a deluge. 
The small mass of comets precludes all danger of this sort. 
Besides, the ocean would require some time before its in- 
ertia would be overcome ; and meanwhile the rapid motion 
of the comet and the rotation of the earth would have pre- 
sented a different surface of water to the comet. 

§ 172. But though proximity is not alarming, it is 
very different with actual contact. The risk of actual con- 
tact is infinitely small when we consider the immense extent 
of the planetary spaces. Still collision is possible, and its 
consequences not without interest to us. If the comet and 
planet were both moving toward the same quarter of the 
heavens each would glide from the surface of the other, 
without any very important change in their movements or 
their physical constitution. But should the directions of 
their respective courses be directly opposite, the conse- 
quences would be far more serious and permanent. The 
inconsiderable mass of the comet would be compensated by 
its prodigious momentum, and the planet might be impeded 
or altogether arrested in its orbit. If the momenta of the 
two bodies were equal, the progressive motion of both 
bodies would be destroyed and they would fall into the 
sun. 

We may perhaps see in the heavens such a collision or 
the consequences of it. The comet of Encke approaches 
nearer to the planets than any other ; it approaches to 
360,000 miles distance of Mercury. This circumstance 
makes a collision between it and Mercury not improbable. 



ELEMENTS OF ASTRONOMY. 117 



CHAPTER IX. 

PHYSICAL ASTRONOMY. 

Analogies observable among the Planets. Their general form and their 
Atmospheres Internal state of our Globe. Central Heat. Theories 
accounting for the external appearance of the Earth and Moon. Objec- 
tions to the theory of Central Heat. Supposed differences of tempera- 
ture in space. Laplace's Nebular Theory. 

§ 173. Let us throw together all we know of the 
physical state of our system and of the fixed stars, and see 
if any light is shed on the former state of the universe. 
We have ascertained that matter exists in the three states 
in which we find it at the surface of the earth. It is solid 
in the planets and moons, and undoubtedly also in the 
nuclei of the sun and stars. It is aeriform at the surface 
of the sun and stars, as is proved by the polarization of 
their light, and also in comets, and in the atmospheres of 
many of the planets, and in ether. It exists in a liquid 
form in several of the planets, as is proved by the clouds 
floating in their atmospheres. But in the moon no water 
is present, and no atmosphere has been detected. 

The atmospheres of the planets differ from one another 
as to color and density. The sun has one or more atmos- 
pheres apparently beside the zodiacal light and ether which 
may in some way belong to him. Of the solid forms of 
matter we know very little. In Mars the red color of the 
soil may be distinctly seen ; a variety of color is also per- 
ceptible on the moon's surface. The shooting stars have 
the same composition as the earth. On the whole, appear- 
ances favor the idea that the composition of the planets is 
identical or only slightly varied. 

. § 174. The general form we have seen is the same 
in all. All likewise depart slightly from this general form. 
Their surfaces are irregular. Mountain peaks are discern- 
ible in the moon, in the nearer planets, and perhaps in the 
sun. Gravity accounts for their general form ; some up- 
heaving force must have caused the departures from it. 



118 ELEMENTS OF ASTRONOMY. 

The upheaving force is much weaker than gravity ; it only 
roughens the surface. 

We call our globe solid and surrounded by a liquid and 
a gaseous envelope. It may also have a liquid interior. 
We know that it increases in density towards the centre. 
But of the state in which the materials of its crust exist 
below a small depth we know nothing. In mines and 
springs the heat of the earth increases about 1° Fahrenheit 
for 54.5 feet. If we suppose the increase to continue in 
an arithmetical ratio, a stratum of granite would be in a 
state of fusion at a depth of twenty-one geographical miles, 
or at between four and five times the elevation of the 
highest summit of the Himalaya. Chemical combinations 
and the neighborhood of volcanoes and the heat imbibed 
from them, may account for some of the internal heat, but 
not for so constant an increase and to so great a depth as 
has been observed in many parts of the solid land. Cen- 
tral heat accounts for it better, and it has therefore been a 
favorite hypothesis. Central heat also accounts for other 
appearances on the surface of the earth. Before we adopt 
it, however, we must see whether no other cause could 
produce the same appearances on the earth, and we must 
seek in the other planets, and particularly in our neighbor 
the moon, for evidence of the existence of central heat there. 
We must also consider whether from what we know of the 
formation of the earth the central portions would be likely 
to remain fluid. 

§ 175. Direct researches on our own globe teach us 
but little. Man's eyes are turned outwards. Accumulat- 
ing all the facts which the best telescopes reveal, with re- 
gard to the distant stars which, strange to say, seem to be 
undergoing more changes than the humble members of our 
system, scrutinizing the planets and especially the moon, 
we may come to some definite conclusion respecting the 
interior of our globe, and the identity of the materials of 
the suns and planets. Nay we may form some idea of the 
circumstances under whose control they took their present 
shape, and may judge which are the older, which the 
newer, inhabitants of the heavens. 



ELEMENTS OF ASTRONOMY. 119 

It is absolutely impossible to explain by central heat 
the present appearance of the moon's surface. In the 
moon the upheaving force is in a far greater ratio to 
gravity than in the earth, hence the mountains of the moon 
bear a much larger proportion to her size than those of the 
earth to the earth's size. If the upheaval was caused by 
the outbreak of a central fluid mass in one planet, it probably 
was in all others. A central fluid opening through cracks 
running nearly in great circles of the earth accounts very 
well for the Ancles and the Himalaya, and other terrestrial 
chains, but accounts not at all for the lunar mountains, 
which are very differently disposed. 

Let us consider whether local heat may not account 
more satisfactorily for the mountains both of the earth and 
the moon. Local heat may be generated in the crust of the 
earth, by water soaking through and coming in contact 
with matter in which it excites violent chemical action. 
The materials once combining under the immense pressure 
of the rocks above, all their gases being kept in, a heat 
would be caused sufficient to melt the hardest substances. 
A large lake of subterranean lava would be formed which 
would uplift plains, throw up mountains, and at length 
vent itself in a volcano. 

§ 176. Another theory supposes that the subterranean 
lakes of lava are remains of the fluid world. Without 
pretending to know what the interior of the earth and moon 
is we may suppose that for ten miles from the surface their 
crust is chiefly solid. We will allow the moon to be in the 
earliest stage of formation, as this suits better her present 
state. Suppose each orb contains in its solid crust liquid 
and gaseous patches, as some crystals contain water. In 
early stages the crust would bubble all over like yeast or 
dough. These bubbles breaking would form cavities like 
the moon's hollows. When all the more external and 
smaller bubbles of the crust have broken, periods of com- 
parative rest would ensue. The earth may now be in this 
state. The melted masses that now lie deeper may some- 
times be chemically disturbed and cause earthquakes and 
volcanoes. The earth has some traces of the state in which 
the moon is. The trap rocks, those which have been in- 



120 ELEMENTS OF ASTRONOMY. 

jected from below into the cracks of other rocks, appear to 
run from centres. In like manner from centres in the 
moon proceed rays which have been called lava, and sup- 
posed to be poured out from centres. This cannot be the 
case because they pass over hill and valley, appearing on 
the sides and at the bottom of precipices like dykes. Per- 
haps this theory explains as well as any other the great 
number of peaks and hollows in the moon and the circular 
form of the few chains discovered there. 

§ 177. The supposition that the interior of the earth 
is a fluid heated mass, agrees well with the nebular theory 
that the earth was once a heated mass of vapor. It is not 
however required by it. If the earth was originally fluid 
it might become solid by either of two modes, from cooling 
or from pressure. The heat would be continually dissipated 
from the surface, and would therefore be greatest at the 
centre ; and so long as the mass was fluid, the inequality 
of the heat would cause a constant circulation between the 
surface and the centre. Now, if the effect of heat in pre- 
venting solidification were greater than the effect of pressure 
in promoting it, solidification would begin at the surface, 
where a crust would be formed, and would constantly in- 
crease in thickness, by layer after layer added to its under- 
side. But if the effect of pressure in promoting solidifica- 
tion were greater than the effect of heat in preventing it, 
solidification would begin at the centre and extend out- 
wardly. While the process was going on, circulation would 
continue in the fluid part exterior to the solid nucleus. 
But before the last portions became solid, a state of imper- 
fect fluidity would arise just sufficient to prevent circula- 
tion. The cooled particles at the surface being then no 
longer able to descend, a crust would be formed, from 
which the process of solidification would proceed far more 
rapidly downwards than upwards from the solid nucleus. 
Our globe would thus arrive at a state in which it would 
be composed of a solid exterior shell, and a solid central 
nucleus, with matter in a state of fusion between them. 

§ 178. If the earth when first thrown off, or when it 
first began to take form, had been suddenly transported 
into regions very much colder than those it left, the outer 



ELEMENTS OP ASTRONOMY. 121 

crust might have been solidified rapidly, and have thus 
imprisoned the melted mass. But from the arrangement 
of the solar system it appears improbable that any such 
violent change of temperature took place. 

But if the solidification was owing to pressure as much 
or more than to change of temperature, it would begin at 
the centre and extend gradually to the surface. The pres- 
sure of gravity would begin to act as soon as the mass was 
insulated. It would vary from nothing at the surface to a 
pressure probably surpassing 100,000 the pressure of our 
present atmosphere. This pressure would reduce all the 
layers of vapor to the solid state, beginning with the cen- 
tral masses, and proceeding toward the surface till nothing 
remained unsolidified but our sea and air. 

§ 179. This change would not be instantaneous, for 
time would be required for each bed to be pushed toward 
the centre. Radiation of heat is so extremely rapid that 
the beds or concentric strata of the earth would get rid of 
the heat developed during their solidification, by radiating 
it through the upper beds still in a vaporous form. So 
that there need not be supposed to remain now, or at any 
definite point of past time, the slightest trace of that heat, 
how great soever it may have been. An effect similar to 
that now described would ensue, in the case of a cylinder 
of great length closed at both ends and filled with the vapor 
of water at the maximum density corresponding to the ex- 
ternal temperature. Were the cylinder horizontal, the 
weight of the fluid could have no influence ; but if it were 
raised up and placed vertically on one of its ends, its. 
weight would produce a pressure on its different beds, in- 
creasing from the top downwards ; and through effect of 
this pressure, liquefaction would take place, beginning on 
the lowest part of the tube and proceeding upwards. The 
time occupied by each bed or stratum in descending would 
not be easy to determine ; but certainly it would sufiice to 
permit the latent heat, developed by the bed liquified im- 
mediately before, to escape by radiating ; and thus the 
water evolved from the vapor would not be heated, but 
would simply preserve the temperature of the external 
space. 

11 



122 ELEMENTS OF ASTRONOMY. 

§ 180. There is one great objection to the theory of 
central heat. We cannot but suppose that some of it would 
be all the time escaping. If so, in the course of ages the 
earth would become smaller. The velocity of the earth's 
rotation depends on her volume. Since, therefore, by the 
gradual cooling of the mass by radiation, the axis of rota- 
tion would become shorter, such decrease of temperature 
would be accompanied by increased velocity of rotation, 
and diminished length of day. Now the comparison of the 
secular inequalities in the moon's motion with eclipses ob- 
served by the ancients, shows that during an interval of 
two thousand years, the length of the day has not been 
diminished by the one hundredth part of a second. We 
know therefore that the mean temperature of the earth 
has not altered during that period so much as the -g-^-g-th 
part of a degree of Fahrenheit. 

§ 181. Abandoning therefore central heat as the 
cause of the elevated temperature of deep pits, two other 
explanations of it have been offered, founded on causes 
whose existence is probable, and which are capable of pro- 
ducing the observed effect. The heat of the different re- 
gions traversed by the earth, while moving with the sun 
and his system through space is unequal. The tempera- 
ture of any part of space may be modified by others having 
specific temperatures, or it may be owing only to the ra- 
diant heat emanating from the different stars, and crossing 
it in all directions. How much the temperature of different 
parts of space varies, from being more or less thickly 
studded with stars, or how much the stars themselves may 
vary as to their heat-giving power, we know not. But we 
may fairly suppose that differences must exist, sufficient to 
account for the evidences of change of temperature we find 
in the earth. Through the extent of the earth's annual 
orbit the temperature of space shows no change. In order 
to perceive it, points at vast distances from one another 
must be compared. But the solar system occupies vast in- 
tervals of time in passing from one of these points to 
another. The changes from one region to the other are 
probably gradual. We could not therefore in a few years 
or perhaps centuries expect to perceive any decided change 



ELEMENTS OF ASTRONOMY. 123 

in our general climate. But if we found that the earth 
retained beneath its surface a store of heat, we should con- 
clude that in passing from a hot to a cold region, on ac- 
count of its massiveness, it had retained a portion of heat. 

§ 182. Now the earth presents indications of having 
been both warmer and colder than it now is. Fossil plants 
and animals are now found in high latitudes where living 
ones of similar character could not exist. To account for 
this we must have passed through a hotter region than the 
present. It must be acknowledged however that the hypothe- 
sis of central heat and of the gradual cooling of the earth 
through millions of years accounts equally well for this. 
But we have also indications that large portions of the 
northern hemisphere, (the same hemisphere in which fossil 
elephants and tropical plants are found,) were once covered 
with glaciers. The increase of cold which is required to 
account for the extensive formation of glaciers is only 14°, 
a quantity which might perhaps be accounted for by our 
passing from the neighborhood of an orb like Sirius to a 
desolate region. Yet after all our speculations it must be 
acknowledged that of no subject are we more ignorant than 
about the globe on which we live. Man's eyes are turned 
outward. Perhaps we are to learn ourselves from others. 
Perhaps the moon or some sister sphere will give us that 
knowledge of the interior of our globe which we vainly 
seek by penetrating its surface. The solution of the prob- 
lem may be withheld from man while he lives on it, or the 
truth may be gradually evolved by the laying aside of error. 
Meanwhile both the astronomer and geologist must contri- 
bute to its further elucidation. 

§ 183. A theory was proposed by Laplace to account 
for the successive changes in the heavens, and to open the 
page of history from the creation of the universe. 

His theory supposed a nebulous fluid to float in space, 
and from this suns and worlds to be gradually evolved and 
thrown off. It rested chiefly on the presence of large 
nebulous masses which were supposed to be slowly formed 
into suns and planets. 

Lord Rosse's telescopes have resolved most of the nebulae 
into stars, and thus rendered doubtful the existence of 



124 ELEMENTS OP ASTRONOMY. 

nebulous matter. But besides that the existence of the 
nebulosity, on which this splendid hypothesis has been 
reared, may reasonably be doubted, the difficulties which 
present themselves in comparing it with the actual state 
of the solar system, are numerous and grave enough 
to warrant the assertion that the name of the illus- 
trious proposer of the nebular theory is after all its chief 
recommendation. So greatly, however, has it influenced 
the modern view of the formation of the solar and sidereal 
systems, that some knowledge of its main features is indis- 
pensable to a right understanding of much that has been 
written on that subject. The following exposition of La- 
place's Cosmogony, given by Pontecoulant, is therefore 
here introduced. 

§ 184. It seems a great deal to have discovered the 
true laws of the celestial motions, and to have been able 
to assign, with so much probability that it almost amounts 
to certainty, the cause which produces them. It might 
have been thought that man ought to rest here ; but in 
science, as in every other thing, success more frequently 
rouses ambition than satisfies it. Master of the great secret 
employed in nature to give life to the planetary system, 
this glory has not sufficed him ; he has sought, by going 
back to past ages, to pierce even the mystery of its forma- 
tion, and he has dared to conceive the bold thought of be- 
ing present, if we may say so, at the spectacle of the cre- 
ation of the world. 

Buffon was the first to start this vast question, and to 
consider from this philosophical point of view the constitu- 
tion of the universe. His ideas on the primitive formation 
of the planets and satellites would find few supporters 
among astronomers now. He supposes that the force of a 
comet falling obliquely on the sun, has projected to a dis- 
tance a torrent of the matter of which it is composed, as a 
stone thrown into a basin causes the water which it contains 
to gush out. This torrent of matter, in a state of fusion, 
has broken into several parts, which have been arrested at 
different distances from the sun, according to their density 
or the impetus they received ; they then united in spheres, 
by the effect of motion of rotation, and, condensing by 



ELEMENTS OE ASTRONOMY. 125 

cold, have become opaque and solid, and formed planets 
and satellites. 

§ 185. This system explains very simply the unity of 
direction of the orbitual motion of the planets, which all 
circulate from west to east, round the sun ; but it is not 
the same with the rotary motion, and we can find no reason 
why this motion should be in the direction of the orbitual 
motion rather than in the contrary one. 

This identity of the direction of the rotary and orbitual 
motions of the planets and satellites, is one of the most 
remarkable facts pointed out by observation. We see no 
better why the orbits of the planets are all nearly circular 
and comprised in a narrow zone of the celestial sphere, 
whilst the comets move in orbits very eccentric, and with 
any inclination whatever to the ecliptic. The hypothesis 
of Buffon is thus very far from explaining the principal 
phenomena which characterize the planetary system, and 
cannot now merit a serious examination. 

§ 186. Let us try, with Laplace, to find out their true 
cause in another way. This new system has for its support 
the labors of Sir W. Herschel, aided by his powerful tele- 
scopes, in regard to the nehnlce, by which appellation those 
whitish spots, seen in different parts of the heavens, of 
which they occupy a large extent, have been called. Ob- 
serving these spots attentively, the nebulous matter is at 
first seen in a most diffused state, and reflecting only a 
feeble and almost uniform light ; in others this matter is 
condensed round one or several dim parts ; in others these 
centres are more brilliant, in proportion to the nebulous 
matter surrounding them ; afterwards the atmosphere of 
each body separating, by an ulterior condensation, there 
result numerous nebulas, formed of brilliant bodies near 
each other, and each surrounded by an atmosphere ; at 
last, a higher degree of condensation changes these nebulae 
into stars. Classing together these observations on a great 
number of different nebulas, Herschel supposes that they 
represent a series of operations on a single mass of nebu- 
lous matter, which would pass from its first state of com- 
pletely diffused, scarcely luminous nebulosity, to the state 
of the most brilliant stars. The progress of the condensa- 
11* 



126 ELEMENTS OF ASTRONOMY. 

tion effected by this change, could only become perceptible 
after the lapse of centuries ; but we may discover it by 
examining at once the whole of the nebulae diffused through 
the sky, as a naturalist, who wishes to discover the suc- 
cessive developements of the organs of an animated being, 
studies them in individuals of different ages. 

Since the attentive observation of the nebulae seems to 
show their change into stars, at epochs more or less re- 
mote, we must suppose from analogy that the existing s-tars 
and the sun himself were formerly masses of nebulous 
matter, reduced by condensation to the state in which we 
now see them. From this induction we are led to regard 
the sun, at the origin of things, as composed of a body 
more or less brilliant, surrounded by a vast atmosphere, 
which extended at first, by the effect of excessive heat, 
beyond the orbits of all the planets, and was confined suc- 
cessively by condensation to its actual limits. 

§ 187. The atmosphere, which we may suppose pos- 
sessed of a rotary motion round its centre of gravity, 
whether this motion results from the reciprocal attraction 
of all its parts, or has been communicated to it primarily, 
must, in condensing by cold, leave in the plane of its 
equator zones of vapor composed of substances which re- 
quired an intense degree of cold to return to a liquid or 
solid state. These zones must have begun by circulating 
round the sun in the form of concentric rings, the most 
volatile molecules of which have formed the superior part, 
;and the most condensed the inferior part. If all the nebu- 
lous molecules of which these rings are composed, had con- 
tinued to cool without disuniting, they would have ended 
hy forming a liquid or solid ring. But the regular con- 
stitution which all parts of the ring would require for that, 
and which they would have needed to preserve whilst cool- 
ing, would make this phenomenon extremely rare. Accord- 
ingly the solar system presents but one instance of this, 
that of the rings of Saturn. Generally the ring must have 
broken into several parts, which have continued to circu- 
late round the sun, and with almost equal velocity, whilst 
at the same time, in consequence of their separation, they 
would acquire a rotary motion round their respective 



ELEMENTS OF ASTRONOMY. 127 

centre of gravity ; and as the molecules of the superior 
part of the ring, that is to say, those farthest from the 
centre of the sun, had necessarily an absolute velocity 
greater than the molecules of the inferior part which is 
nearest it, the rotary motion, common to all the fragments, 
must always have been in the same direction as the or- 
bitual motion. 

§ 188. However, if after their division one of these 
fragments has been sufficiently superior to the others to 
unite them to it by its attraction, they will have formed 
only a mass of vapor, which, by the continual friction of 
all its parts, must have assumed the form of a spheroid 
flattened at the poles and elongated in the direction of its 
equator. Here then are rings of vapor left by the succes- 
sive retreats of the atmosphere of the sun, changed into so 
many planets in the condition of vapor circulating round 
the sun, and possessing a rotary motion in the direction of 
their revolution. This must have been the most common 
case ; but that in which the fragments of some ring would 
form several distinct planets possessing different degrees of 
velocity, must also have taken place, and the four tele- 
scopic planets, Ceres, Juno, Pallas and Vesta, discovered 
at the beginning of the present century, seem to present 
an instance of this ; at least, if it is not admitted, with 
Olbers, that they are the fragments of a single planet, 
broken by a strong interior commotion. It is easy to 
imagine the successive changes produced by cooling on the 
planets whose formation has just been pointed out. Indeed, 
each of these planets, in the condition of vapor, is, in every 
respect, like one of the nebulse in the first stage ; each 
must, therefore, before arriving at a state of solidity, pass 
through all the stages of change we have just traced in the 
sun. 

§ 189. At first, the condensation of its atmosphere 
will form round the centre of the planet a body composed 
of layers of unequal density, the densest matter having, by 
its weight, approached the centre, and the most volatile 
reached the surface, as we see in a vessel different liquids 
ranged one above another, according to their specific grav- 
ity, arrive at a state of equilibrium. The atmosphere 



128 ELEMENTS OF ASTRONOMY. 

of each planet will, like that of the sun, leave behind it 
zones of vapor, which will form one or several secondary 
planets, circulating round the principal planet, as the moon 
does round the earth, and the satellites round Jupiter, 
Saturn and Uranus, or else they will form, by cooling 
without dividing, a solid and continuous circle, of which we 
have an instance in the rings of Saturn. In every case, 
the direction of the rotary and orbitual motion of the satel- 
lites or the ring, will be the same as that of the rotatory 
motion of the planet ; and this is completely confirmed by 
observation. 

§ 190. The wonderful coincidence of all the planetary 
motions, a phenomenon which we cannot, without infringing 
the laws of probability, regard as merely the effect of 
chance, must then be the result of the formation of the 
solar system in this ingenious hypothesis ; we see also 
why the orbits of the planets and satellites are so little 
eccentric, and deviate so little from the plane of the solar 
equator. A perfect harmony between the density and 
temperature of their molecules in a state of vapor, would 
have rendered the orbits rigorously circular and made them 
coincide with the plane of this equator ; but this regularity 
could not exist in all parts of such large masses ; there have 
resulted the slight eccentricities of the orbits of the planets 
and satellites, and their deviation from the plane of the 
solar equator. 

When in the zones, abandoned by the solar atmosphere, 
there are found molecules too volatile either to unite with 
each other or with the planets, they must continue to re- 
volve round the sun without offering any sensible resistance 
to the motions of the planetary bodies, either on account of 
their extreme rarity, or because their motion is affected in 
the same way as that of the bodies they encounter. These 
wandering molecules must thus present all the appearances 
of the zodiacal light. 

§191. We have seen that the figure of the heavenly 
bodies was the necessary result of their fluidity at the be- 
ginning of time. The singular phenomenon presented by 
the rigorous equality indicated by observation, among the 
lesser motions of rotation and revolution of each satellite, 



ELEMENTS OP ASTRONOMY. 129 

an equality rendering the opposed hemisphere of the moon 
forever invisible to us, is another obvious consequence of 
this hypothesis. Indeed, supposing that the slightest dif- 
ference had existed between the mean motion of rotation 
and revolution of our satellite whilst it was in the state of 
vapor or of fluidity, the attraction of the earth would have 
elongated the lunar spheroid in the direction of its axis to- 
wards the earth. The same attraction would have tended 
to diminish insensibly the difference between the rotary 
and orbitual motions of the moon, so as to confine it to 
narrow limits. 

§ 192. The principal phenomena of the planetary sys- 
tem are therefore explained with great facility by the hy- 
pothesis we are examining ; and as these successive changes 
of a nebulous mass, and the leaving of a part of its sub- 
stance by cooling, agree with all the leading phenomena, 
it must be allowed a high degree of probability. In this 
hypothesis the formation of the planets would not have 
been simultaneous ; they have been created successively 
at intervals of ages ; the oldest are those which are farthest 
from the sun, and the satellites are of a more recent date 
than their respective planets. It may be, if we are ever 
permitted to reach so high, that by an examination of the 
constitution of each planet, we may go back to the epoch 
of its formation, and assign to each its place in the chro- 
nology of the universe. It is likewise seen that the velocity 
of the orbitual motion of each planet as it is now, must 
differ little from that of the rotary motion of the sun, at 
the period when the planet was detached from its atmos- 
phere. And as the rotary motion is accelerated in pro- 
portion as the solar molecules are confined by cooling, so 
that the sum of the areas which they describe round the 
centre of gravity would remain always the same, it follows 
that revolutionary motion must be so much more rapid as 
the planet is nearer the sun; and this is seen by observation. 

§ 193. It likewise results that the duration of the ro- 
tation either of the sun or of a planet, must be shorter than 
the duration of revolution of the nearest body which circu- 
lates round them ; this observation is completely confirmed 
even in those cases where the difference between the dura- 



130 ELEMENTS OP ASTRONOMY. 

tion of the two motions must be very slight. Thus the in- 
terior ring of Saturn being very close to the planet, the 
duration of its rotation must be almost equal, but a little 
longer than that of the planet. The observations of Herschel 
give indeed 0.432 of a day as the duration of the rotation of 
the ring, and 0.427 of a day as that of the planet ; why then 
should w T e not admit that this ring has been formed by the 
condensation of the atmosphere of Saturn, which formerly 
extended to it ? We may perhaps deduce from the laws of 
mechanics, and the actual dimensions of the sun, and the 
known duration of its rotation, the relation existing be- 
tween the radius vector of its surface and the time of its 
rotation in the different stages of condensation through 
which it has passed. The third law of Kepler would be no 
longer the mere result of observation ; it would be directly 
deduced from the primordial laws of the heavenly bodies. 

§ 194. In this system, as in that of BufFon, the par- 
ticular form of the planets, the flattening at the poles, and 
bulging out at the equator, is only the necessary conse- 
quence of the laws of the equilibrium of fluids, and easily 
explains the greater part of the phenomena observed by 
geologists in the constitution of the terrestrial globe, which 
appear inexplicable, if it is not admitted that the earth and 
planets have been originally fluid. 

Let us now see what is the origin and part assigned to 
comets by this hypothesis. Laplace supposes that they do 
not belong to the planetary system, and he regards them 
as masses of vapor formed by the agglomeration of the 
luminous matter diffused in all parts of the universe, and 
wandering by chance in the various solar systems. Comets 
would thus be, in relation to the planetary system, what 
the aerolites are in relation to the earth, with which they 
seem to have no original connection. When a comet ap- 
proaches sufficiently near the regions of space occupied by 
our system, to enter into the sphere of the sun's influence, 
the attraction of that luminary, combined with the velocity 
acquired by the comet, causes it to describe an elliptic or 
hyperbolic orbit. But as the direction of this velocity is 
quite arbitrary, comets must move in every direction and 
in every part of the sky. 



ELEMENTS OF ASTRONOMY. 131 

§ 195. The cometary orbits will then have every in- 
clination to the ecliptic ; and this hypothesis explains 
equally well the great eccentricity by which they are 
usually affected. Indeed, if the curves described by comets 
are ellipses, they must be greatly elongated, since their 
major axes are at least equal to the radius of the sphere of 
the sun's attraction ; and we must consequently be able to 
see only those whose eccentricity is very great and perihe- 
lion distance inconsiderable ; all others,- on account of their 
minuteness and distance, must always be invisible ; unless, 
at least, the resistance of the ether, the attraction of the 
planets, or other unknown causes, diminish their perihelion 
distance, and bring them nearer the terrestrial orbit. The 
same circumstances may change the primitive orbits of 
some comets into ellipses whose major axes are compara- 
tively small ; and this has probably happened to the pe- 
riodical comets of 1759, 1819 and 1832. The laws of 
curvilinear motion likewise show that the eccentricity of 
the orbit chiefly depends on the direction of the comet's 
motion on its entering the sphere of the sun's attraction ; 
and as this motion is possible in every direction, there are 
no limits to the eccentricities of the orbits of comets. 

If, at the formation of the planets, some comets pene- 
trated the atmospheres of the sun and planets, the resist- 
ance they met would gradually destroy their velocity ; they 
would fall on those bodies describing spirals, and their fall 
would have the effect of causing the planes of the orbits 
and equators of the planets to remove from the plane of 
the solar equator. It is, therefore, partly to this cause, 
and partly to those we have developed above, that the 
slight deviations Ave now perceive must be attributed. 

§ 196. Such is the summary of the theory of Laplace, 
on the origin of the solar system. This hypothesis explains, 
in the most satisfactory manner, the three most remarkable 
phenomena presented by the planetary motions. 

" 1st. The motion of the planets in the same direction, 
and nearly in the same plane. 

2d. The motion of the satellites in the same direction 
as their planets. 

3d. The singular coincidence in direction of the rota- 



132 ELEMENTS OF ASTRONOMY. 

ry and orbitual motions of the planets and the sun, which 
in other systems would present inexplicable difficulties. 

The no less remarkable phenomena of the smallness of 
the eccentricities and inclinations of the planetary orbits 
are also a necessary consequence of it, whilst we see, at the 
same time why the orbits of the comets depart from this 
general law, and may be very eccentric, and have any in- 
clination whatever to the ecliptic. The flattening of the 
form of the planets, shown on the earth by the enlargement 
of degrees of the meridian, and by the regular increase of 
weight in going from the equator to the poles, is only the 
result of the attraction of their molecules whilst they were 
yet in a state of vapor, combined with the centrifugal force 
produced by the rotary motion impressed on the fluid mass. 
In short, among the phenomena presented by the motions 
and the form of the heavenly bodies, there are none which 
cannot be explained with extreme facility by the successive 
condensation of the solar system ; and the more this system 
is examined, the more we are led to acknowledge its proba- 
bility. 

§ 197. Undoubtedly, if, as Laplace has himself said, 
a hypothesis not founded on observation or calculation must 
always be presented with extreme diffidence, this, it will be 
granted, acquires, by the union and agreement of so many 
different facts, all the marks of probability. But what 
principally distinguishes it from the ordinary theories con- 
cerning the formation of systems, is the identity which it 
establishes between the solar system and the stars spread 
so profusely through the sky. 

All the phenomena of nature are connected, all flow 
from a few simple and general laws, and the task of the 
man of genius consists in discovering those secret connec- 
tions, those unknown relations, which connect the phenom- 
ena which appear to the vulgar to have no analogy. In 
going from a phenomenon of which the primitive law is 
easily perceived, to another in which particular circum- 
stances complicate it so as to conceal it from us, he sees 
them all flowing from the same source, and the secret of 
nature becomes his profession. Thus the laws of the ellip- 
tic motion of the planets led Newton to the great principle 



ELEMENTS OP ASTRONOMY. 133 

of universal gravitation, which he would have sought for 
in vain in the less simple phenomena of the rotary motion 
of the earth, or the flux and reflux of the sea. 

§ 198. But, this great principle being once discovered, 
all the circumstances of the planetary motions were ex- 
plained, even in their minutest details, and the stability of 
the solar system was itself only the necessary consequence 
of its conformation, without which, as Newton thought, 
God would be constantly obliged to retouch his work, in 
order to render it secure. Laplace, extending to all the 
stars, and consequently to the sun, the mode of condensa- 
tion by which the nebula are changed into stars, has con- 
nected the origin of the planetary system with the primor- 
dial laws of motion, without recurring to any hypothesis 
but that of attraction. He has, therefore, extended the 
great law of universal gravitation, which is. probably the 
only efficient principle of the creation of the physical 
world, as it is of its preservation. 

The hypothesis of Bufibn required not only the fall on. 
the sun of a comet as large as the mass of all the planets 
and satellites, which is very improbable, but in order to 
explain the formation of the innumerable planetary systems 
which the imagination may conceive round each star, as; 
round the sun, it would have been necessary to imagine 
the fall of so many new comets, that reason would soon re- 
fuse to believe in chances so often repeated, and always 
when there was need of them. The principle of the con- 
densation of the atmosphere of the nebulae is, on the con- 
trary, general, and would produce phenomena nearly anal- 
ogous, in all the stars and planets. 

12 



134 ELEMENTS OF ASTRONOMY. 



CHAPTER X 



THE ROTATION AND FIGURE OE THE EARTH. 

Effect of the Motion of the Earth on the apparent Motion of the Heavens. 
Distinction between the Earth's yearly and diurnal Motion. Axis of 
Rotation. The Circles of the Equator and Ecliptic. Shape of the Earth's 
Orbit. Invariability of the Earth's Rotation. Earth's Figure and Dimen- 
sions. Its Ellipticity. Pendulum experiments. Measurement of Arcs of 
a Degree on the Earth's surface. 

§ 199. The reader will remember those groups of 
islands which stud the Pacific. On the map they are near 
neighbors, but in fact they are too far apart to be visible to 
one another, and are separated by thousands of miles t>f 
ocean. Thus the inhabitants of each group might easily 
suppose their own to be the only group of the Pacific. And 
as the islands of each group are likewise widely separated 
from one another, it would not be strange if each island 
should be deemed by its inhabitants a world in itself. 

Let us choose from a group containing some hundreds 
of islands, an island not precisely in the centre of the 
group. And let us suppose it endowed with a magnetic 
power like the Black Mountain of fairy tales. We will 
suppose several boats of different sizes to circulate round it 
at distances varying from 3 to 190 yards. Let each boat 
also twirl continually round an imaginary axis, while it pro- 
ceeds in its course round the island. Place upon this boat 
a tiny insect, whose life endures only some seventy revolu- 
tions round the island, and this insect, supposing it endowed 
with reason, will have greater facilities for learning the 
wide Pacific with all its isles than man has for becoming ac- 
quainted with the universe. The earth is our boat, which 
measures from end to end 8,000 miles or upwards of forty- 
two millions of feet. The island round which we circle is 
our sun, and it is so bright, that from the part of the boat 
toward him we can discern no other body in the universe. 
But when our side of the boat is turned from him we see 
other boats, the planets our companions, and we also see 



ELEMENTS OF ASTRONOMY. 135 

the myriad isles of our cluster, beaming fixed stars in the 
heavens. 

§ 200. Since the difficulties in the way of learning 
are so great we will begin modestly and study our own 
planet first. There are three questions which must be set- 
tled before we can take the earth as our stand-point, and 
our measure for other bodies. The first is whether it is at 
rest or in motion ; the second, what is its shape ; the third, 
what is its size ? 

On looking into the heavens we find nothing apparently 
at rest. The sun is not in the same place if we look at 
him after an interval of an hour ; the moon, if she is visible 
in the heavens, changes her place with rapidity. The stars 
which at twilight we saw in the east are over our head at 
midnight, and at dawn set in the west. Not only this, but 
the sun no two consecutive days rises or sets in the same 
place ; and a different set of stars make their appearance 
in the east each night. Yenus appears sometimes as an 
evening, sometimes as a morning star. The other planets 
are often absent from our heavens, 

§ 201. The first division we should attempt to make 
of these motions is by finding which are common to all, and 
which are peculiar to some. Those which are common to all 
may arise from either of two causes, and we must choose the 
more probable of the two. Either all the objects we look 
at are moving or we may ourselves be in motion, An ob- 
server in a vessel on a wide ocean cannot distinguish 
whether another vessel within sight is sailing and his own 
at rest., or whether the other vessel is at rest and his own 
drifted along by some unsuspected current. An observer 
on one ship cannot judge accurately the rate of progress or 
the direction of the other. For supposing them both in 
parallel directions, the swiftest will appear to advance with 
only the difference of their velocities. Suppose them sail- 
ing in opposite directions, the other will recede with the 
sum of their velocities, If they are sailing in diverging or 
converging directions from one another, the apparent direc- 
tion will not for one moment coincide with the true one. 

But instead of being on the ocean let the observer be on 
a lake whose banks are covered with villages and varied bv 



136 ELEMENTS OF ASTRONOMY. 

woods and hills. If he finds that all these objects pass in 
the same direction and at an equal rate from his sight, and 
that in a certain time he has seen them on every side of 
him, will he not more rationally ascribe the motion to his 
single vessel than to the whole landscape around him. 
And if he sees on the lake other ships, not joining in this 
motion of the shores, but outstripping or lagging behind, 
he will not hesitate to ascribe to these ships proper mo- 
tions ; remembering always that he does not see these mo- 
tions as they really are, but altered by his own motion. 

§ 202. Thus when we are obliged to choose between 
the simultaneous motions of many bodies, or the motion of 
one body, we keep on the side of probability when we at- 
tribute the motion to the single body. 

Applying this principle to our earth, it is much more 
probable that by turning round she is a part of the twenty- 
four hours exposed to the sun, and another part in the 
presence of the stars, than that the sun and all the stars 
sweep round her in one uniform diurnal motion. It is much 
more likely that she is so inclined in her orbit that in one 
season of the year she presents a given part of her surface 
but a short time, and in another season a longer time to the 
sun, than that he, varying his motion, describes now lower 
and shorter, and again higher and longer arcs in the 
heavens. 

We are therefore warranted in considering the earth's 
real motion as the cause of all the apparent motions which 
are common to other bodies. And we may ascribe to the 
same cause that part of each body's apparent motion for 
which it will account. Thus we find in the heavens three 
kinds of motion, one purely apparent caused by the earth's 
real motion, one made up of the body's proper motion 
combined with the earth's motion, and one among the fixed 
stars too distant to be affected by the earth's motion, and 
for all we know a simple proper motion. 

§ 203. Let us now consider what motions the earth 
must have to account for all the appearances we observe. 
There are two sets of phenomena to be accounted for, the 
diurnal and the yearly ; we must therefore allow her two 
motions. We find that the sun appears to pass through an 



ELEMENTS OF ASTRONOMY. 137 

entire circuit of the heavens in the course of the year, 
moving from west to east. This apparent path of the sun 
may be caused by a real motion of the earth in the same 
direction from west to east. For supposing the sun to be 
motionless and the earth to be moving round him, the in- 
habitants of the earth would refer the sun to a point in the 
concave surface of the sphere precisely 180° from the 
earth's place. If the earth moved 15° eastward, the sun 
would move 15° eastward ; if in a year the earth passed 
through the whole circle, the sun would appear to pass 
through the whole circle, always remaining precisely in 
opposite quarters of the heavens to those in which the 
earth would be. All these phenomena would be precisely 
the same and would occur in the same order whether the 
sun moves round us or we move round the sun. 

We have now to account for the daily rising and setting 
of the sun and stars. To do this we need only suppose in 
the earth rotation in the same direction with its revolution. 
As the earth rotating toward the east turns up a portion of 
her surface toward the sun, the sun appears to that portion 
to rise in the east. As the place comes under the direct 
beams of the sun it is noon at the place ; as it turns further 
round, it leaves the sun behind, he becomes invisible, but 
the stars rise ; the earth still turns round, the place is 180° 
away from the sun, it is midnight ; the earth turns another 
quarter, the place has left the stars behind, they have set ; 
it has come in sight of the sun r the sun has again risen. 
This motion of rotation is confirmed even by phenomena on 
the surface of the earth. A stone let fall from the top of a 
high tower, falls not at the foot of the tower, but a little 
further east, showing that being further from the earth's 
centre it had a greater velocity than the earth it fell on. 

§ 204. But while the earth has been turning it has 
passed on in its orbit, it refers the sun to a different part 
of the concave sphere, the sun rises among stars lying far- 
ther east than those among which it rose yesterday. The 
stars which rose after sunset the night before are now lost 
in his beams, those which rose later now appear as soon as 
he is out of sight. Those which were on the meridian at 
12* 



138 ELEMENTS OF ASTRONOMY. 

midnight now pass it before midnight, and new stars come 
into view before the sunrise. 

The daily and yearly changes may be exhibited in minia- 
ture by a bright light placed in the centre of a room whose 
walls are covered with tapers. Let the light represent the 
sun, the tapers the fixed stars. By walking round this 
room and rotating once every minute we shall face the 
lamp and thus make it rise to us once every minute ; we 
shall also have the tapers facing us a half of every minute. 
If we walk round the lamp so slowly that we can turn 
round 365 times before we return to our starting point, we 
shall perform a course similar to that of the earth during 
a year. For by our revolution round the lamp we shall 
have brought it in succession between ourselves and every 
part of the wall, while by our rotation we shall have 
brought the light facing us, that is we shall have had 365 
noons, and 365 times we shall have had a partially different 
set of tapers exhibited to us in each rotation. 

§ 205. If the earth rotates it must have an axis or 
line of immovable particles passing through its centre ; and 
the diameter of this axis, and the fact of its variability or 
invariability we can only ascertain from the heavens. An 
axis may be variable in its position within the sphere that 
is as to the place where its poles touch the surface, it may 
be invariable with regard to the solid sphere, and may 
vary in direction carrying along the sphere with it, or it 
may be invariable in both respects. In the latter case it 
will always point toward one point, and all surrounding ob- 
jects will appear to be carried in circles round that one 
point. As we find one point in the northern celestial 
hemisphere and another in the southern, round which all 
the stars appear to describe circles, we have no hesitation 
in saying that the axis of our earth is immovable, and if 
prolonged would pass through these points. On the earth's 
surface and in a line with those two immovable points in 
the heavens are two points which do not rotate, and par- 
take only of the orbitual motion of the earth. These are 
the north and south poles of the earth. 

The equator is an imaginary great circle on the earth's 
surface 90° from each pole, dividing the earth's surface 



ELEMENTS OF ASTRONOMY. 139 

into two hemispheres. Its plane is perpendicular to the 
earth's axis. 

The latitude of a place on the earth is its distance from 
the equator north or south. The polar distance or angular 
distance from the nearest pole is the complement of the 
latitude. 

The intersection of the plane of the equator with the 
sphere of the heavens is called the equinoctial. The dis- 
tance of a star north or south of the equinoctial is called 
its declination. 

§ 206. The secondaries to the equator are called 
meridians because that secondary which passes through the 
zenith of any place is called the meridian of that place, and 
is at right angles both to the equator and the horizon, pass- 
ing as it does through the poles of both. 

In the heavens these secondaries are called hour circles 
because the arcs of the equinoctial intercepted between 
them are used as measures of time. 

Longitude on the earth's surface is reckoned from some 
arbitrary point. The English and Americans reckon from 
the observatory at Greenwich, 180° westward, and 180° 
east longitude. It is more convenient to reckon 360° 
westward. 

. As declination corresponds to latitude so does right as- 
cension in the heavens correspond to terrestrial longitude. 
It is likewise reckoned from a point arbitrarily chosen, in 
the vernal equinox, where the equator cuts the earth's 
path in spring, the beginning of our northern year. If the 
star is situated in the equator its right ascension is the 
number of degrees of the equator between the star and the 
vernal equinox. But if the star is north or south of the 
equator, then its right ascension is the arc of the equator 
intercepted between the vernal equinox and that secondary 
to the equator which passes through the star. 
. § 207. The sun's apparent and the earth's real path 
as marked out among the stars is an (almost) invariable 
great circle, and is called the ecliptic. One half of it is 
north and the other half south of the equator. Its plane 
is inclined 23° 28' to that of the equinoctial, consequently 
its poles lie 23° 28' distant from those of the equinoctial. 



140 ELEMENTS OE ASTRONOMY. 

Celestial latitude and longitude are to the ecliptic what 
declination and right ascension are to the equinoctial ; and 
both longitude and right ascension are reckoned from the 
same point, the vernal equinox. 

The latitude and longitude of the heavenly bodies are 
not observed directly on account of the difficulty of verify- 
ing the proper instruments intended for this purpose. 
When required they are deduced by calculation from the 
observed declination and right ascension. All the apparent 
motions caused by the earth's rotation are referred to the 
equinoctial and its secondaries immediately. All apparent 
motions arising from the earth's revolution are more con- 
veniently referred to the ecliptic. 

§ 208. The axis of the earth is always parallel to 
itself and always apparently points near the polar star. 
For the diameter of the earth's orbit is as nothing in com- 
parison with the distance of the fixed stars. Two parallel 
bars a few yards asunder both apparently point to the 
moon when in the horizon. And these two or three yards 
may bear to 240,000 miles, the moon's distance, a greater 
ratio, than 190,000,000 of miles do to our distance from 
the polar star. Thus in all parts of its orbit the axis of the 
earth points in the same direction, and from the immense 
distance of the star to which we refer it, compared with 
the small size of the earth's orbit, it appears to point to the 
same place. Thus the pole of the heavens is the vanishing 
point not only of all great circles perpendicular to the 
equator in one position of the globe but at all seasons of the 
year. North to a person at the equator is in a line per- 
pendicular to the equator ; but all these lines referred to 
the distant concave sphere unite and we call north a point. 
In the same way all the hour circles which could be drawn 
at all seasons of the year vanish in two opposite points of 
the equinoctial. 

As all planes perpendicular to the equator vanish in the 
poles, so do all planes parallel to the equator vanish in the 
equinoctial ; for the polar diameter of the earth can cause 
displacement in the heavens no more than the equatorial 
diameter. An observer at each pole and one at the equa- 
tor will therefore see the equinoctial in the same place. 



ELEMENTS OE ASTRONOMY. 141 

All planes parallel to the ecliptic vanish in the ecliptic ; 
all planes perpendicular to it vanish in the poles of the 
ecliptic. 

Thus the motions of the earth furnish us with two sets of 
circles by means of which we can determine the position of 
the heavenly bodies. We learn the ecliptic directly, and 
infer the place of its poles. We observe the position of the 
poles of the earth, and infer the place of the equator from 
them. 

§ 209. We must not however suppose that because 
the line which the earth describes in the sphere in a year 
is a circle, that its orbit is itself precisely circular. Its 
true form may be ascertained by measuring the sun's ap- 
parent diameter ; or by observing his angular velocity at 
different parts of the year, and ascertaining whether this 
is such as would take place in a circular or in an elliptic 
orbit. 

If the diameter of the sun be accurately measured it is 
found to be greater at some periods of the year than at 
others. If such observations be continued for several 
successive years, it will still further appear, that, at the 
same time in each year, his diameter will be equal to what 
it was the year before. We must then conclude, either 
that the sun regularly expands and contracts, or that our 
distance from him is variable. The former supposition is 
absurd. We must therefore adopt the latter. If the earth 
remained at the same distance from the sun at all periods 
of the year, that is, were its orbit circular, the sun's diam- 
eter would never vary. Since the sun's diameter does 
vary we are persuaded that its orbit is not a circle. And 
since the angle under which an object is viewed, varies in- 
versely as the distance, the variation in the sun's diameter 
will enable us to determine what the form of the orbit is. 
If at the second observation, the diameter is half what it 
was at the first, the second distance will be double the 
first ; if one third, the distance will be treble ; and so on, 
If we begin on a particular day to measure the sun's diam= 
eter, and for every week for a whole year continue to do 
the same, and if we set off on paper lines radiating from a 
common centre, proportioned to these different measure- 



142 ELEMENTS OF ASTRONOMY. 

ments, by joining the extremities of those lines, we shall 
accurately represent the form of the earth's orbit. 

§ 2 1 0. For instance, draw a straight line and take in it a 
point, Plate I. Fig. 4, for the position of the sun. On the 
21st of December the sun's diameter is 32£'- its maximum ; 
on the 21st of June it is about 31 J' its minimum. Since the 
distance is inversely as the apparent magnitude, we will set 
off towards I, that part of the orbit where the earth is in 
our winter, from a convenient scale of 63 equal parts, 
(that being the number of half minutes- in 31 J'.) We will 
also set off from the same scale 65 equal parts, (65 being 
the number of half minutes in 32^',) measuring them on 
the same straight line running through S, but on the part 
of it between S and the part of the earth's orbit occupied 
by it in summer. When the year has advanced six weeks, 
or one eighth of the whole, let the sun's diameter measure 
32f. This indicates the proportional distance of the earth 
six weeks before or after mid summer. Draw a line from 
the sun in a direction determined by the amount of the 
earth's angular motion in six weeks at that season, and 
measure off on it 64| equal parts. When a quarter of a 
year has elapsed the diameter will be 31 T y, and the line 
must contain 63^ equal parts, its direction being known 
from the angular motion of the earth, J>y increasing the 
number of these observations, and joining the extremities 
of the proportional lines thus laid down, we shall have an 
ellipse with the sun in one of the foci, In the figure the 
eccentricity is much greater than is actually the case in 
the earth's orbit. 

In learning the point of the earth's orbit in which the 
sun's apparent diameter is largest we have discovered its 
perihelion or point nearest the sun. In learning the point 
in which the sun appeared smallest we have determined 
the earth's aphelion or point farthest from the sun. These 
being known we may ascertain the amount of the eccen* 
tricity of the earth's orbit. The sun's greatest apparent 
diameter is 32' 35", 6 ; its least is 31' 31 /; ; hence the ra* 
dius vector at the aphelion ; radius vector at the perihe- 
lion ; : 32.5933 : 31.5167 : : 1.032 : 1. Half of the di£ 
ference of the two equals the distance of the focus of the 



ELEMENTS OE ASTRONOMY. 14S 

ellipse from the centre, a quantity which is the measure of 
the eccentricity of a planetary orbit. 

§ 211. As we have learned the variations of the 
earth's distance from the sun by noting the sun's varying 
size, so we may learn the varying speed of the earth's mo- 
tion in her orbit by observing the variations in the sun's 
apparent motion. From what was said before it is evident 
that whatever motion in her orbit the earth has we attribute 
to the sun. If therefore the sun at the perihelion moves 
in twenty-four hours over an arc of 61', while at the aphe- 
lion he describes in the same time an arc of only 57', we 
must suppose these changes to take place in the earth's 
motion. Our varying distance from the sun accounts for 
a portion of these changes, for the angular velocity varies 
inversely as the distance. If it accounts for all then the 
rate between the largest and shortest arc described in 
twenty-four hours would be the same as that of the largest 
and smallest apparent diameters. That is §£=1.07, would 
equal ff-ifftf =1-034. But the first fraction is the square 
of the second, for 1.07 = (1.034) 2 . Therefore the earth's 
angular velocities are to each other inversely as the squares 
of the distance at the perihelion and the aphelion. And 
this is found to be true of every part of the orbit. 

If the angular velocities described by the earth in single 
days are inversely as the squares of the distances, the dis- 
tances are inversely as the square roots of the arcs. Thus 
the relative distance of the earth from the sun in every 
point of its revolution may be easily calculated. Thus its 
perihelion distance is to its aphelion as \^51 to V61, or as 
1 : 1.034. The difference between the two is nearly ^V of 
the perihelion distance, a quantity, as we shall see hereaf- 
ter, no less than 3,000,000 miles. This variation of the 
earth's speed in her orbit thus discovered from observation 
is what we shall find must take place in elliptic orbits, and 
thus confirms the form of orbit which we have assigned 
to it. 

§ 212. In the earth's daily rotation we find no varia- 
tion. Equal arcs of the equinoctial present themselves in 
equal times, proving that the earth's rotation is always 
equally rapid. If there were any jerk or sudden stoppage 



144 ELEMENTS OF ASTRONOMY. 

in the motions of the earth it would be perceptible to us 
by its effects. Even a diminution of her rotary speed 
■would precipitate the Atlantic on the shore of Europe and 
lay bare the eastern shores of America. But no such vio- 
lence is seen in the heavens. The impulse was from the 
beginning, and given by an all-powerful hand. All on the 
surface of the earth partakes its motion, and we are no 
more sensible of it than the fly who goes back and forth on 
the deck of a vessel is conscious of his motion down the 
stream. All around us seems at rest or moving only by 
its own motion. The bird who hovers over a field shares 
the motion of the air, or he would appear to us to move 
toward the west. Thus every motion we make is in reality 
combined with the rotary and revolving motion of the 
earth, the translation of the sun, and perhaps with the pro- 
gression of our whole cluster, and even with other grand 
movements. 

§ 213. We will now inquire into the shape of the 
earth. Many circumstances obvious to the most careless 
observer give us- an idea of its form. We will begin with 
these, and then from more exact experiments and reasoning 
deduce its precise form. 

In the other planets and the moon we see only the glob- 
ular form, we may therefore expect from analogy to find 
the earth a globe. The earth's shadow cast on the moon 
in an eclipse is always convex. Eclipses take place when 
the earth is in every variety of position between the sun 
and moon, and a body which whatever way it is turned 
casts a circular shadow must be a globe or sphere. A cone 
or a cylinder may cast a circular shadow when the sun is 
in the direction of its axis, but a shadow which is circular 
on whatever side it is projected must come from a sphere. 
When on the ocean a ship comes in sight, at first only the 
top of her mast and gradually the whole mast and hull rise 
to view ; and as the ship sails away the hull, mast and 
yards successively disappear. As we approach land the 
mountains, steeples, houses, and lastly the shore itself be- 
comes visible ; as we sail away, they gradually sink away 
at the base. This may be seen in all parts of the ocean, 
because owing to the mobility of water the parts of the 



ELEMENTS OE ASTRONOMY. 145 

globe covered with water keep their normal shape. It may 
be seen in extensive plains or deserts ; but in most parts 
the surface of the earth is so rugged or so interrupted by 
minor objects that we are less sensible of these phenomena. 
This irregularity, however, is too slight to affect the general 
surface of the globe ; the highest mountains are but five 
miles above the level of the sea, and their mass no more 
takes off from the earth's rotundity than particles of dust 
from the form of a celestial globe. These phenomena 
occur every where and terrestrial objects rise and vanish with 
equal rapidity in whatever direction we approach or leave 
them ; thus proving that the earth has a convex surface 
which falls away equally or nearly equally in all direc- 
tions. 

§ 214. Hence in constructing canals an allowance of 
eight inches a mile below the horizontal plane is necessary, 
to compensate for the curvature of water at rest. And if 
we imagine a large portion of the ocean which now appears 
a plane surface to be frozen, and to be cut off, a slice which 
is two miles across would rise eight inches in the middle.. 
Rivers flow to the sea because their origin is above the 
normal curve of the earth. They indicate by the slowness 
or rapidity of their course the slope of the country they 
traverse, that is the excess of their curve over the curve- 
of the sea. 

Another proof of the earth's globular form is the testimony 
of those who have sailed round her. She is of moderate 
size and has been circumnavigated in more than one direc- 
tion, and navigators assure us that they have every where 
observed the gradual appearance and disappearance of ob- 
jects, and every where their view has been bounded by 
a circumference about three miles distant. This extent 
included within the visible horizon is a slice of the earth, 
and the reason no greater extent is visible is that a globe 
the size of the earth falls away so much that it is out of 
sight beyond this bounding line. On a smaller globe the 
surface would fall away more and the terrestrial horizon 
would be less extensive, on a larger globe we should have a 
wider horizon. 

13 



146 ELEMENTS OE ASTRONOMY. 

Now the only body from which none but circular sections 
can be made is a sphere ; the earth therefore must be a 
sphere or nearly a sphere. 

§ 215. Before inquiring into the exact form of the 
earth we must make ourselves acquainted with another set 
of circles and poles which like the ecliptic and equatorial 
with their secondaries, serves to define the place of a star. 
I speak of the celestial horizon, a circle every where 90° 
distant from the zenith or pole over our heads and from 
the nadir or pole beneath our feet. This great circle is 
not immovable and the same to all the inhabitants of the 
globe like the equinoctial and ecliptic, it differs with the 
position of each observer. We make our own horizon. 
By advancing north or south, east or west, on the globe, 
we change the point over our heads and consequently the 
boundary line 90° distant from that point. If we stand 
still, the earth in her rotation presents us each moment 
with a new horizon, and each night, owing to her revolu- 
tion, she presents us a horizon partially different from the 
one we saw at the same time the night before. It is often 
more convenient to refer heavenly bodies to the horizon 
and its secondaries then to the equinoctial. The observed 
distances can afterward be converted into the declination 
and right ascension or the celestial latitude and longitude. 
The distance of a star above the horizon is called its alti- 
tude ; it is measured on a vertical circle. The body may 
also be referred to the zenith by a vertical circle, and the 
arc intercepted is called its zenith distance. The zenith 
distance is the complement of the altitude. 

§ 216. Azimuth is the angular distance of a celestial 
object from the north or south point of the horizon (ac- 
cording as it is the north or south pole which is elevated), 
when the object is referred to the horizon by a vertical 
circle. Or it is the angle comprised between two vertical 
planes, one passing through the elevated pole, the other 
through the object. When the body is on the horizon it is 
only necessary to count the number of degrees between 
that point and the meridian in order to find its azimuth. 
But if the point is above the horizon, its azimuth is esti- 
mated by passing a vertical circle through it and reckoning 



ELEMENTS OF ASTRONOMY. 147 

the azimuth from the point where this circle cuts the hori- 
zon. The altitude and azimuth of an object being known, 
its place in the visible heavens is determined. 

We may learn the zenith and nadir by direct observa- 
tion, or we may learn the horizon and find the zenith and 
nadir, each 90° distant from it. The nadir is always in 
the direction of a plumb line, the zenith 180° from the 
nadir. The zenith and nadir are the vanishing points of 
all lines in all parts of the earth mathematically parallel to 
the direction of a plumb line at the observer's station. 

The celestial horizon is the vanishing line of a system of 
planes parallel to one another, passing, one through the 
centre of the earth, another through the place of the spec- 
tator, another through the point on the earth's surface op- 
posite to him. All these planes cut the concave sphere in 
one and the same line, for the earth included between them 
is no more to them than a grain of sand between two circles 
of paper. The edges of the paper coincide and so do all 
the parallel planes which can be drawn between one ex- 
tremity of the earth and the other, if they are only con- 
tinued sufficiently far. 

§ 217. This great circle thus marked out in the 
heavens, and passing through the centre of the earth is 
called the celestial or rational horizon. It divides the 
sphere into two equal parts. But as we do not stand at 
the centre of the earth we use the plane parallel to this 
and touching the earth where we stand, called the sensible 
horizon. The sensible horizon is parallel to the rational, 
and distant from it by the earth's radius, or 4,000 miles. 
As we have above said, it coincides with the rational hori- 
zon, of course it divides the sphere into two equal parts. 
The rising and setting of stars are the same to the sensible 
and the rational horizon. The sun rises a very little and 
the moon a little earlier, and both set later to the ra- 
tional than to the sensible horizon. The horizon or bound- 
ing line which to our eye unites earth and heaven is a dif- 
ferent circle from these. It is not a great circle of the 
earth, and consequently when prolonged not a great circle 
of the heavens. 

An eye perfectly level with the surface of the sea would 



148 ELEMENTS OF ASTRONOMY. 

have no visible terrestrial horizon, and would see that portion 
of the heavens above the sensible horizon. Let the eye 
be raised even to the height of a man, five feet, and a seg- 
ment of the earth's surface, nearly three miles in every 
direction becomes visible. The bounding line of the seg- 
ment is our visible horizon. It is not in the same plane 
with the sensible horizon, but below it. When prolonged 
it cuts the concave sphere below the sensible and rational 
horizon. This depression of the visible horizon below the 
direction of a spirit level is called the clip of the horizon. 
It must always be allowed for, as observers are always 
more or less raised above the level of the sea. To get the 
true altitude of a star above the sensible horizon, the angle 
of depression must be subtracted from the observed alti- 
tude. 

§ 218. If the earth were a larger sphere, a portion 
of its surface would fall away less from a plane, and the 
dip of the horizon would be less. On a smaller globe the 
depression would be greater, and more of the heavens 
below the sensible horizon would be brought into view. 
Knowing the amount of the dip, the falling away of the 
earth in a given portion of her circumference, and conse- 
quently the convexity of her surface, her size may be cal- 
culated ; for a given convexity can belong only to a sphere 
of a certain size. 

As we ascend elevations, though a much greater ex- 
panse of country is exposed to our view, not only is each 
object smaller but the whole picture appears smaller, be- 
cause it is included within a smaller angle. This may be 
seen in an instant by opening a pair of dividers so as to 
embrace part of a globe. When they are nearly wide 
open, they will include a very small portion of the globe, 
but the eye of the spectator at the centre of the dividers 
would see this portion under a very large angle. If the 
dividers are closed more and gradually lifted, they will 
embrace more of the globe, but the visible portion will be 
seen under a smaller angle. In like manner a movable 
tangent passes from the eye of the spectator in every direc- 
tion forming his horizon. As he ascends, this tangent 
reaches to a greater distance on the globe, but forms with 
itself a smaller angle. 



ELEMENTS OE ASTRONOMY. 149 

§ 219. Let us now consider what effect elevation has 
on this visible plane considered as a circle of the heavens. 
If the earth were cut in halves and a man stood in the 
centre of its plane surface his height would be so trifling 
compared to the surface on which he stands, that his visible 
and rational horizon would agree. The sun, moon and 
stars would rise over the rational and visible horizon at the 
same moment. But as man stands on a pinnacle formed 
by the convexity of the earth, it is the same as if part of 
the earth were hewn from under him and no longer kept 
out of sight so much of the heavens. The more it is hewn 
away, that is the more convex it becomes, the more of the 
starry sphere is visible, for here, as in the moral world, 
only the opaque earth hides the heavens. In considering 
this effect of the earth's convexity it is better to compare 
the visible with the sensible than with the rational horizon ; 
the effect of the earth's convexity in increasing the visible 
portion of the heavens will be more apparent than if we 
use the rational horizon. "We see its effect daily in the 
lighting up of mountain tops long before and after valleys 
are shadowed. Aeronauts tell us they have seen the sun 
set three or more times in one evening. This was occa- 
sioned by repeatedly increasing their elevation, and thus 
again bringing the sun above their horizon. On the Peak 
of Teneriffe, a mountain 13,000 feet high, Humboldt found 
the surface of the sea depressed on all sides nearly two 
degrees. The sun rose to him twelve minutes sooner than 
to an inhabitant of the plain ; and from the plain the top 
of the mountain was seen enlightened twelve minutes be- 
fore the rising and after the setting of the sun. 

§ 220. The dip of the horizon at different elevations 
may be observed ; or the shape and dimensions of the earth 
being known, it may be calculated, for different elevations. 
When the spectator's eye is one foot above the sea-level, 
the dip is 59", when 100 feet, it is 9' 51". As almost all 
land is raised above the level of the sea, and as observa- 
tions are usually taken in high places, tables have been 
constructed showing the dip of the horizon at all required 
elevations. 

The reason that the observer sees only the segment of 
13* 



150 ELEMENTS Of ASTRONOMY. 

the earth's surface which is bounded by the visible horizon 
is that the earth falls away so suddenly that the edge of 
what is seen conceals all beyond. If the earth were a 
larger sphere a given amount of elevation would make a 
larger ring-shaped surface visible than now becomes so ; 
if it were a smaller sphere, the same amount of elevation 
would bring up a narrower ring below the natural horizon. 
As it is the visible horizon enlarges rapidly as the observer 
mounts. An eye placed five feet above the surface of the 
sea sees 2f miles every way. If it be elevated twenty 
feet, that is to four times the height, it will see 5£ miles, 
or twice the distance. At a height of 100 feet the horizon 
is 13 miles off. 

A much larger portion of the atmosphere than of the 
earth is visible to us. The greatest distance of the clouds 
in the horizon at sea is 94 miles in every direction from an 
observer. Consequently the whole extent or diameter of 
the horizon is 188 miles ; and the circumference is 590.97 
miles. Thus the physical visible horizon extends only 2f 
miles ; if we look higher it extends 94 miles ; if we take in 
the fixed stars it is millions of millions of miles. 

Probably the greatest extent of surface even seen at 
once by man was in the aeronautic expedition of Biot and 
Guy Lussac. They were elevated nearly five miles above 
the surface of the earth. Now the convex surface of a 
spherical segment is to the whole surface of the sphere to 
which it belongs as the versed sine or thickness of the seg- 
ment is to the diameter of the sphere. Its thickness in 
this case almost exactly equals the perpendicular elevation 
of the point of sight above the surface. The proportion of 
the visible area to the whole earth's surface would in this 
case be that of five miles to 8,000 or of one to 1,600. 
The portion visible from Mount Etna or the Peak of Tene- 
riffe is about one four thousandth. 

Having found rudely the shape of the earth, we will, 
before we attempt to ascertain its precise shape, learn its 
actual size. This is of the utmost importance, for it is one 
■unit in calculating the size and distance of the heavenly 
bodies. As we cannot embrace in one view any large por- 
tion of the earth, nor can we retire to a distance and view 



ELEMENTS OF ASTRONOMY. 151 

it as a whole, all the knowledge we can have of it is gained 
by exact measurements of small portions of its surface ac* 
companied by geometrical deductions. If we were sure 
the earth was a sphere we could at once tell its precise 
dimensions. If we can learn that it is a spheroid, the pro- 
cess will be somewhat more difficult. 

§ 221. We may ascertain the earth's diameter by 
measuring the heights and distances of two stationary points 
which can barely be discerned from each other, and but 
for the effect of refraction, by which we are enabled to see 
a little round the interposed segment of the earth's surface, 
this method would be pretty correct. It is known from 
observation that two points each ten feet above the earth's 
surface are visible from each other over still water at a 
distance of nearly eight miles. This, by a simple calcula- 
tion, gives 8,450 miles for the diameter of the earth. 

Another method, which is the one usually adopted, 
is to measure the length of a degree of the meridian. 
Thus let the latitude of the place g be determined with 
great accuracy by repeated observations of the heavenly 
bodies ; determine also the latitude of h ; for simplifying 
the process, we will suppose the two places to be exactly 
north and south of each other. If the distance between 
them be measured with great care, a simple proportion will 
give the approximate circumference of the globe ; thus, 
supposing the latitude of h to be 50° 54' N., and that of g 
to be 56° 24' N., and the measured distance between g 
and h to be 390 miles. Then 5° 30' : 360° : : 390 miles : 
24,900 miles =the circumference of the globe. 

§ 222. TVe may learn the shape of the earth by 
studying phenomena at its surface, or by considering 
whether the attraction of the earth on the moon is that of a 
sphere or a spheroid. 

The simplest though not the most exact way which pre- 
sents itself is to ascertain the direction of the surface as it 
Would exist without the accidental irregularities of hills and 
mountains ; and this is determined by means of a plumb 
line, the line always taking a direction perpendicular to the 
surface. The plumb line is to be sure slightly diverted 
from the centre of the earth by the attraction of mountains, 



152 ELEMENTS OF ASTRONOMY. 

and sometimes by dense subterraneous matter to a greater 
degree ; but this difficulty may be obviated by multiplying 
experiments. 

If the earth is a perfect sphere plumb lines let fall on all 
portions of its surface will converge to one point. If not 
they will converge along certain curves in the interior of 
the earth. Three methods have been employed to investi- 
gate the curvature of the earth's surface. It has been 
inferred from the measurement of degrees in different 
places, from the vibrations of a pendulum, and from certain 
inequalities in the moon's orbit. The first method is di- 
rectly geometrical and astronomical. In the two others 
we infer from movements accurately observed, the measure 
of the forces which cause those movements ; and from the 
inequality of the forces we infer the difference between the 
equatorial and polar diameters. Conclusions respecting 
the figure of the earth, founded on the increase of the at- 
tracting force in going from the equator to the poles, are 
dependent on the distribution of the density of the inte- 
rior. Its ellipticity may be calculated on the supposition 
that the earth does or does not increase in density toward 
the centre. A comparison of the earth's figure with its 
velocity of rotation makes an increase of density toward 
the centre probable, and a comparison of the ratios of the 
polar and equatorial axes of Jupiter and Saturn with their 
times of rotation shows the same increase to exist also in 
these planets. Actual measurements calculated on the 
supposition that there is this central density, give very 
nearly the same compression as the measurement of degrees 
and the moon's inequalities. 

§ 223. If the earth is not a perfect sphere some por- 
tions of its surface must be nearer to the centre than others, 
consequently gravity must act more strongly in these por- 
tions than elsewhere. The most delicate measure of the 
force of gravity which we have is the swinging of a pendu- 
lum, because this shows on a large scale the effects of grav- 
ity. A pendulum is a weight suspended either by a line 
or by a bar of wood or metal. If the weight is drawn aside 
from the perpendicular, gravity draws it back to that line, 
and the momentum thus acquired causes it to describe an 



ELEMENTS OF ASTRONOMY. 153 

arc on the other side of the perpendicular. If the arc of 
vibration is small the times of vibration are equal as long 
as the motion continues. 

There is a constant relation between the lengths of pen- 
dulums and the times of their vibrations ; the same rela- 
tion which exists between the spaces and the times of 
bodies falling with accelerated motion. The squares of the 
times of vibration are in proportion to the lengths of the 
pendulums. 

If two pendulums be made of the same length and one 
be carried to the equator, the other near the pole, and it 
is found that one of these vibrates more rapidly than the 
other, we must suppose it influenced by a more powerful 
attraction. At the equator the vibration of a pendulum 
is slackened, showing that gravitation is weaker there ; 
near the poles vibrations are more rapid, showing increased 
gravitation. We infer therefore that we are nearer the 
centre at the poles than at the equator, and that the earth 
is not a sphere but a spheroid. By comparing pendulums 
of the same length, and marking the number of vibrations 
made by them in equal times in different latitudes, the 
amount of gravitation in these places is ascertained. The 
intensities of the forces will be as the squares of the num- 
bers of vibrations at the two places. A pendulum which 
under the equator makes 86,400 vibrations in a mean solar 
day, transported to London, makes 86,535 vibrations in 
the same time. Hence gravitation at the equator as to 
that at London as 86,400 2 : 86,535 3 , or as 1 : 1,00315. 

§ 224, Experiments made in all accessible latitudes 
give T ^% for the difference in gravitation at the equator 
and at the poles ; the weight increasing (from causes we 
shall presently explain) as the square of the sine of the 
latitude. 

Since the weight of all bodies is affected in the same 
proportion it is not easy to prove this increase of gravita- 
tion directly. The weights we would use are as much in- 
creased toward the poles as that which we would balance 
with them. What we call a pound at the equator, is at 
the pole more than a pound, though we still call it a pound, 
The pressure exerted on the hand by a mass weighing 194 



154 ELEMENTS OF ASTRONOMY. 

pounds at the equator, would be at the poles equal to the 
pressure of a mass weighing 195 pounds at the equator. 
This has been thus explained. Imagine a weight x sus- 
pended at the equator by a string without weight passing 
over a pulley and conducted, if such a thing were possible, 
over other pulleys till the other end hangs down at the 
pole and there sustains the weight y. If then the weights 
x and y are such as at either station, equatorial or polar, 
would exactly balance one another, they would not in this 
supposed situation balance one another, but the polar 
weight would preponderate ; to restore the balance the 
weight x must be increased by the T ^ ¥ of its mass. 

§ 225. Let us consider whether all the increased 
gravitation shown by the pendulum as we approach the 
pole is owing to the increased convexity of the earth, or 
whether some other cause may exist. The rotation of the 
earth immediately suggests one, in the centrifugal force 
created by it. This centrifugal force is greatest at the 
equator, because the equatorial particles revolve with more 
rapidity than any others. Moreover the direction of the 
centrifugal force communicated by rotation is always oppo- 
site to the direction of the radius of the different circles in 
which the particles move. ISTow gravity or the centripetal 
force is always in a direction perpendicular to the surface, 
and therefore it is only at the equator that this is exactly 
in an opposite direction to the centrifugal force. The 
quantity of force directed from the surface (for strictly 
speaking it should not be called centrifugal force) at any 
point not in the equator may be resolved into two forces, 
only one of which is directly opposed to gravity, and is the 
true centrifugal force. The other is perpendicular to the 
direction of gravity, and does not diminish its force. Thus 
only in the equatorial regions does the whole amount of 
centrifugal force take effect in diminishing the force of 
gravity. In parts distant from the equator a portion only 
of the centrifugal force, and that in its diminished state, 
acts in opposition to the force of gravity. The diminution 
of gravity at the surface of the earth arising from the 
centrifugal force varies as the square of the cosine of the 
latitude. 



ELEMENTS OE ASTRONOMY* 155 

§ 226. This centrifugal force at the equator lightens 
bodies by -zfe part of what they would otherwise weigh 
there, and by ^|~f of what they would weigh at the pole 
were the earth a perfect sphere. At the poles the gravita- 
tion of a body equals the whole force of gravity at that dis- 
tance from the centre, at the equator it equals the force 
of gravity at that distance minus the centrifugal force. 
We must therefore, in accounting for the variations of the 
pendulum, attribute one part of the change to the unequal 
convexity of the earth, and one part to the inequality of 
centrifugal force in different latitudes. Of these two causes 
of diminution the centrifugal force is the more powerful- 
It diminishes gravitation by 2ib-> while the elliptic form of 
the earth diminishes it only O . The two together make 
up the difference shown by the pendulum - x fa. The earth's 
size and velocity of rotation being known, her ellipticity 
and the consequent increase of gravitation at the poles, 
and also the diminution of gravitation at the equator, aris- 
ing from the centrifugal force, are matter of pure calcula- 
tion. Since the effects calculated for those two causes 
equal the observed difference the assigned causes are the 
true ones. 

The rate of increase of gravitation shown by the pendu- 
lum, in travelling from the equator to the pole, is as the 
square of the sine of the latitude. Gravitation would be 
inversely according to the squares of the radii if the earth 
were at rest. It is diminished in each latitude by centri- 
fugal force. Centrifugal force in each latitude, as we have 
shown, is according to the squares of the cosines ; there- 
fore the actual gravitation in each latitude is as the square 
of the radius minus the square of the cosine, that is, as the 
square of the sine. 

The earth's polar diameter is to its equatorial as 298 : 
299 ; or more correctly the polar diameter is 7,899.171, 
and the equatorial 7,925.648 miles. For the convenience 
of round numbers it is usually called 8,000 miles. The 
difference between the polar and equatorial radii is thirteen 
miles. The excess of the equatorial radius is 4f the height 
of Mount Blanc, or 2 J the probable height of the Dhawa- 
lagiri, in the Himalaya chain. 



156 ELEMENTS OF ASTRONOMY. 

§ 227. We will now see what information concerning 
the earth's shape and size we can obtain from the heavens. 
Since the earth affords us no marks bj which to trace or 
estimate our course we must look to the stars to ascertain 
our progress and to guide our path. The polar distances 
of the stars are known ; therefore by observing their meri- 
dian altitudes we can learn the height of the pole and con- 
sequently our own latitude. When our latitude has dimin- 
ished a degree we know that, provided we have kept to the 
meridian and the earth is a symmetrical figure, we have 
described one three hundred and sixtieth part of the earth's 
circumference. By observations on the polar star we may 
keep the true direction of the meridian even if local diffi- 
culties oblige us to turn aside from it for a while. Owing 
to the irregularity of the earth's surface, which often 
makes it impossible to travel due north or south or east or 
west, and which constantly changes the plane of the ob- 
server, it is exceedingly difficult to determine the length of 
a degree of a great circle of the earth. Of course it is not 
necessary to measure a whole meridian, a few degrees in 
known latitudes are sufficient, and calculation will then 
give the true form of the whole. Neither is it necessary 
to measure precisely a degree, we need only* know exactly 
how much, be it more or less, we have measured. We de- 
termine by astronomical observations the exact difference of 
the latitude of the two stations, and measure this distance 
on the ground accurately. It is of great importance to 
avoid even slight errors in these two operations, for an 
error committed in a single degree will be magnified 360 
times in the circumference and nearly 115 times in the 
diameter of the earth concluded from it. The altitude of the 
observed star also, since it affects the latitudes, must be 
found very accurately. The true place of stars which are 
near the zenith can be found more precisely than the place 
of those which are nearer the horizon. Instead therefore 
of observing stars which may be nearer the horizon, a star 
near the zenith is observed, and if its zenith distance is 
raised or depressed a degree we know that the pole must 
also have been raised or depressed a degree, for the polar 
distance of the star cannot have altered. If then at one 



ELEMENTS OF ASTRONOMY. 157 

station a star passes through the zenith and at another it 
passes one degree north or south of the zenith, we are 
sure that the latitude of the places differ by the same 
amount. 

§ 228. If all measurements of a degree on all great 
circles of the globe agreed we should have no hesitation in 
calling the earth a sphere. If in travelling in one direc- 
tion round the sphere we should find that equal portions of 
the celestial sphere rise to equal measured distances of 
advance we should infer that the earth is symmetrical in 
that direction. If travelling in a second direction for 
equal advance on earth unequal portions in the heavens- 
rise, we shall infer that in this direction the earth differs 
from a sphere. 

There is but one great circle of the earth, equal por- 
tions of which correspond to equal portions of the celestial 
sphere ; this is the equator. When a man on the equator 
advances sixty geographical miles, his horizon (allowing 
always for the earth's rotation) will advance one degree on 
the celestial sphere. For each of the 360° of longitude he 
will see successively an additional degree of right ascen- 
sion. If a man traverses the earth from north to south in 
the line of a meridian, he no longer finds that for equal 
distances travelled his horizon advances equally. At the 
equator 68.73 statute miles north or south bring up a 
degree of the heavens. In latitude 43°, 68.99 are re- 
quired ; in 80° lat., 69.36 ; and at the poles 69.39 miles 
are required to bring up one degree. The more rapidly a 
surface alters its curve, the more does its tangent vary at 
each observation, the less distance need we go to vary it a 
certain amount, for instance one degree. Comparing ob- 
servations in longitude and in different latitudes we infer 
that the equator is the only great circle of the earth, and 
is larger than any other curve on its surface ; and likewise 
that the curvature of the earth is greatest at the equator, 
and consequently that the meridians, though called circles, 
are in reality elliptic curves ; and are shorter than the 
equator. 

We must apply the same correction to the form of the 
visible actual horizon formerly considered as round. It 
14 



158 ELEMENTS OF ASTRONOMY. 

can be perfectly circular only at the poles. At the equator 
it must be elliptical, and longer from east to west than 
from north to south. Between the poles and the equator 
it assumes various forms composed of oval curves having 
the shorter axis running north and south. 

To determine the exact amount of the earth's ellipticity 
requires nice observations in different latitudes. 

§ 229. More than a dozen measurements of degrees, 
most of which belong to this century, have now made us 
quite accurately acquainted with the dimensions and the 
ellipticity of our globe. Only two of these measurements 
have been throughout their whole extent actual and me- 
chanical. The irregularity of the surface of most coun- 
tries renders this mode inadvisable. The first of these, 
made in 1635, was of that arc of the meridian which lies 
between London and York. The difference of the latitudes 
of these cities was first ascertained. This gave the number 
of degrees in the arc to be measured. The distance be- 
tween the two cities was then actually measured, and the 
turnings and windings of the road, and the ascents and 
descents were afterward allowed for. This measurement 
gave the length of a degree too large by 1,000 yards. 

The only other instance of the actual measurement of an 
arc is that made by Mason and Dixon. They measured 
with rods a line of nearly 100 miles, in Pennsylvania, near 
latitude 39° 12'. 

§ 230. A more accurate mode of finding the length 
of a degree is by a combination of actual measurement, 
and of trigonometrical operations founded upon it. Two 
places are selected which lie under the same meridian or 
nearly so, and the difference of their latitudes, which gives 
the number of degrees of the arc to be measured, is ascer- 
tained with the utmost precision. A base line of a few 
miles of extent, and at some little distance from the meri- 
dian arc is then very carefully measured ; this is the only 
actual measurement which need be made. The extremities 
of this base line are then connected with the extremities 
of the meridian arc by imaginary triangles, the sides of 
which are not measured, but trigonometrically determined 
from the length of the first base line and the angles of the 



ELEMENTS OP ASTRONOMY. 159 

triangles. A theodolite measures the angles accurately. 
The stations from which the angles are observed should be 
so selected that none of the angles shall be very small, 
otherwise a slight mistake in an angle causes a great error 
in the opposite side of the triangle. Toward the conclusion 
of the process one of the sides of one or more of the tri- 
angles is measured and its length compared with that found 
by computation. This base of verification is taken as far 
distant from the first base as circumstances will admit. In 
one of the French operations the base of verification was 
between four and five hundred miles distant from the first 
base, and was seven miles in length, and yet the difference 
between its computed length and that obtained from its 
actual measurement did not amount to twelve inches. 

§ 231. The following particulars show the accuracy 
which distinguishes these operations and the means taken 
to ensure it. A base of five miles in length was measured 
in Hounslow Heath with a steel chain of exquisite work- 
manship. The same base had been measured three years 
before with glass rods, and the two measurements differed 
only 2 J inches. Sometimes rods of platina or of iron are 
used for measuring, and an allowance is made for the 
changes of temperature affecting the rods in the course of 
the operation. In later measurements rods composed of 
different metals put together so as to show the slightest 
contraction or expansion have been used. 

In 1735 two scientific expeditions were sent from France, 
to determine the length of a degree of longitude in different 
latitudes. One degree was to be observed upon the equa- 
tor, the other as near the poles as possible. One degree 
was measured in the valley of the river Tornea in Lap- 
land. The base was measured on the frozen surface of 
the river, with a view to obtain as level a plain as possible ; 
and rods of deal were employed instead of metal on ac- 
count of the extreme cold. Two independent measures by 
two sets of observers differed only four inches. These 
operations were completed several years before the return 
of the Peruvian expedition, which had to contend with ex- 
traordinary difficulties caused by the ill-will and indolence 
of the natives, and by the localities. Their station was a 



160 ELEMENTS OF ASTRONOMY. 

mile and a half above the level of the sea, and in some in- 
stances the heights of two neighboring signals differed more 
than a mile. To accomplish their measurements occupied 
nine years, three of which were employed in the determi- 
nation of latitudes alone. 

At the beginning of the French revolution, a measure- 
ment was made from Dunkirk to Barcelona, in order to as- 
certain the length of a quadrant of a meridian, and take 
the ten millionth part of it as a metre or universal standard. 
A metre contains 39.37 inches. This method of obtaining 
a standard of measure is not so good as the English mode, 
which consists in observing the length of the pendulum, 
which in a certain latitude, (that of London,) in a vacuum 
at the level of the sea beats seconds of mean time. The 
length of the pendulum is ascertainable without the use of 
any linear measure whatever ; whereas in determining the 
French standard or the quadrant of the meridian, some 
linear measure already in use must be employed. And 
thus the very basis of their new system is expressed in 
terms of that in the place of which it is substituted. Arcs 
of longitude have also been measured in various other lati- 
tudes, and their observed lengths agree with their theoret- 
ical lengths. 

§ 232. By measuring degrees of the meridian we ob- 
tain the compression of the earth, while pendulum observa- 
tions give us the ellipticity confounded with the effects of 
centrifugal force. The motions of the moon confirm the 
shape assigned to the earth. They cannot be accounted 
for on the supposition that the earth is a sphere, but they 
agree perfectly with the supposition that it is a spheroid. 
The ellipticity inferred from the lunar inequalities has an 
advantage not possessed either by measurements of degrees 
or by pendulum experiments, in being independent of local 
accidents, and thus showing the mean ellipticity of the 
earth. 



ELEMENTS OF ASTRONOMY. 161 

CHAPTER XI. 

GENERAL PHENOMENA ON THE EARTH'S SURFACE. 

Universal diffusion of Gravity over the Earth's Surface. Determination of 
the Earth's Mass and Density. Our Knowledge of the Earth's Surface. 
The Sea. Tides. Stability of the Ocean's Equilibrium. The Atmos- 
phere. Clouds. "Winds. Trade "Winds. Use of the Atmosphere. 
Absorption and Diffusion of Light and Heat. Refraction. Twilight. 

§ 233. The earth's exact diameter is important as a 
unit of measure for other bodies ; its density is no less so 
as a standard of comparison for other planets. 

The powerful attraction which the earth, in consequence 
of its superior bulk, has for bodies on its surface, prevents 
our perceiving the attraction they exercise on one another. 
It may be shown however by balancing a small mass in 
such a manner that it may be moved by the slightest in- 
fluence, and then bringing a large body into its neighbor- 
hood ; or by ascertaining the deflection of the plumb line 
caused by the vicinity of a mountain ; or by comparing the 
length of a pendulum vibrating seconds in a plain, and on 
the summit of a mountain. A balance of torsion is in fact 
a horizontal pendulum. It may be applied to a mountain 
or to much smaller attracting bodies. If two equal balls 
of lead are suspended from the opposite extremities of a 
slender bar of wood, and this is suspended at its centre by 
a very fine wire, the only force required to move the balls 
will be that which suffices to produce a slight twisting of 
the wire that suspends the rod. Now if a large mass of 
lead be brought into the neighborhood of each ball, (the 
rod having been previously hanging at rest,) its attraction 
will cause the rod to turn round, until the small balls have 
come into the same line with the large masses. If the 
masses be now moved a little further, the balls will follow 
them ; twisting the wire from which the rod is suspended 
still more. Now, as the force which is required to produce 
any amount of alteration in the position of the rod can be 



162 ELEMENTS OE ASTRONOMY. 

ascertained in another way, the actual amount of the at- 
traction which the masses exercise over the balls may be 
determined ; and this may be compared with the earth's 
attraction. From the knowledge of these facts, the quan- 
tity of matter in the earth may be compared with that in 
the masses of lead ; for the weight of the earth is just as 
much greater than that of the masses of lead, as the force 
with which it attracts the balls exceeds that with which the 
masses attract them, proper allowance being made for their 
difference of distance. When the actual weight of the 
earth is known, we may estimate its density as compared 
with water ; since we may easily calculate the weight of a 
globe of water of equal size. And from the weight and 
density of the earth, that of the other planets and of the 
sun may be ascertained. 

§ 234. A beautiful proof that the attraction of gravity 
is diffused through separate portions of the earth is the 
fact that mountains draw a plumb line out of the perpen- 
dicular. 

By ascertaining the exact amount of the deviation and 
obtaining the specific weight of the mountain, the specific 
weight of the earth, or its weight compared with the moun- 
tain and consequently with a globe of water of the same 
size, may be learned. The mountain Schehallien in Scot- 
land was thus examined. It was measured from its base 
to its summit, its component parts examined, and its spe- 
cific gravity determined. Assuming that the spirit in the 
levels of the instruments would be attracted toward the 
mountain, or that the plumb lines by which the instruments 
are rectified would deviate from a perpendicular to the 
horizon, it is plain that observations on the fixed stars, 
taken on opposite sides of the mountain, would differ from 
each other by double the amount of deviation. 

§ 235. The meridian zenith distances of certain stars 
were observed first on the north and then on the south 
side of the mountain. They gave a constant error of 
llj" more than could be accounted for by the difference 
in latitude of the stations. The mountain therefore de- 
flected the plumb lines from the perpendicular 5f". 
From the actual attraction of the mountain its attraction 



ELEMENTS OF ASTRONOMY. 163 

at the distance of the earth's centre was to be calculated. 
The comparative powers of attraction of the earth and the 
mountain, and their relative sizes being known, their rela- 
tive densities could be determined. After a year's labor 
in reducing these data it was found that the density of the 
earth was to that of the mountain as five to three, or nearly 
five times the density of water, or nearly double that of 
rocks near the surface. It is of about the density of silver 
ore throughout. Since the density of the earth is so much 
greater than the average specific gravity of rocks on the 
surface, it follows that the internal matter must be more 
dense than the superficial layers. 

§ 236. While we thus by reasoning learn the size and 
mass of the earth, how little do we actually know of it. 
Even its surface is not yet wholly known to us. The an- 
cients thought a fiery impassable zone separated the north- 
ern from the southern regions ; the polar circles have 
proved equally impenetrable to the most zealous efforts of 
the moderns. The interiors of the continents are yet 
unexplored, so that enough remains to stimulate the curios- 
ity and enterprise of man for ages yet to come. Enough 
of the surface has been explored to prove to us that its 
nature is every where the same. Every where there are 
traces of convulsive change ; every where fire and water 
leave their traces. Huge rocks have been melted and 
cast up ; water has worn them away and left the sand and 
the pebbles to tell of the slow destruction it has wrought. 
To water also we owe all the plains and habitable spots of 
the earth. It is still busy bringing all things to its own 
level. All over the globe there is no sameness, each coun- 
try has its characteristic scenery, every where there is 
variety. Even the bottom of the sea has its risings and 
its abysses, fit abodes now for its varied inhabitants, and 
perhaps the peaks and valleys of a world to be upheaved 
hereafter. But beyond this very outer surface we cannot 
penetrate. The deepest mines are but as a scratch on the 
surface of a model globe, the highest mountains are not 
five miles high. The ocean has not been sounded below 
27,600 feet. The deepest mines do not penetrate more 
than 2,231 English feet, or ■§£$& of the earth's radius be- 



164 ELEMENTS OF ASTRONOMY. 

low the level of the sea. The absolute depth of mines 
often exceeds this, for they are usually situated in elevated 
valleys or mountains ; sometimes the absolute is less than 
the relative depth. By studying the edges of strata which 
dip and rise again at a distance, and by observing their dip 
we can learn with absolute certainty the depth of the basin 
formed by them. We can thus infer the nature of the 
coast six thousand feet below the level of the sea. By 
adding these depths to the mountain summits, we have 
48,000 English feet or -jj^ of the earth's radius known to 
us. Loosely infolding the hollows of this surface lie the 
waters of the ocean. Gravity keeps them in their place, 
gravity swells them and gives them the true form of the 
earth, the normal form which it would retain all over were 
it not for the rigidity of the rocks. 

§ 237. If we would learn the general form of the earth's 
surface we must seek it in the most mobile substance, for 
this obeys no minor principles of stratification, but the great 
forming principle of gravity. The ocean gives us the form 
of the globe when not interfered with by the hardness of 
materials. It also shows us that this form is liable to 
change. Twice every day some parts of the globe's out- 
line increase, others diminish in convexity. 

The moon is so near to the earth that the difference be- 
tween the earth's surface and its centre bears a considera- 
ble ratio to the moon's distance ; it is 4,000 out of 240,000 
miles. The waters next under the moon are therefore as 
much attracted as the centre of the earth and more also. 
It is this excess of the moon's attraction which causes the 
tide. As the sun is so much more distant the earth's ra- 
dius bears to his distance a smaller proportion ; the differ- 
ence between his attraction on the earth's surface and at 
its centre is less. Thus although the sun is so much larger 
than the moon it affects the tides only § as much. 

In obedience to gravity the waters nearest the moon fall 
towards the moon and form tides. The waters nearest the 
moon are of course less distant from the moon than the 
solid earth is ; these particles move easily, they are drawn 
up. The earth also is attracted, but not so much. The 
waters most distant from the moon are still less attracted 



ELEMENTS OF ASTRONOMY. 165 

and they remain more distant. Thus the waters in a 
straight line with the moon bulge from the earth's surface ; 
part of them because they are drawn from the earth, part 
because the earth is drawn from them. Opposite sides of 
the earth always have similar and equal tides at once, 
whether high or low. The sun and moon act in the same 
direction when the moon is new or when it is full. In 
both cases the tides are at their highest, and are called 
spring tides, and occur twice a month. When the moon is 
in her quarters the sun and moon act at right angles to one 
another, the tides are low and are called neap tides. The 
solar wave is lowest when the lunar wave is highest, and 
the solar highest when the lunar lowest. 

§ 238. The point of the earth's surface to which the 
moon is vertical is the highest point of the waters ; 90° in 
every direction from this the waters are drawn away in 
consequence of the moon's attraction and the limited quan- 
tity of water ; and to all such places it is low tide. Since 
every meridian is under the moon and opposite to it in the 
course of twenty-four hours, it is high tide twice and low 
tide twice in twenty-four hours. In those latitudes to 
which the moon is never vertical the tides cannot be so 
high as in those nearer the zodiac. Within the polar cir- 
cles their ebb and flow are scarcely discernible. The 
highest and lowest tides occur in March and September, 
because then the sun and moon are in or near the same 
plane. 

Since the moon advances 13 Q daily in her orbit, the 
earth must not only make a diurnal revolution but advance 
13°, before the midnight moon will be seen on the same 
meridian as on the preceding night. To accomplish this 
takes about fifty minutes. The mean duration of the ebb 
and flow will then be half of 24h. 50', or 12h. 25' ; and in 
one flow of the twenty-four hours the place will be under 
the moon, the next high tide removed from it by a space 
equal to more than the diameter or the semi-circumference 
of the earth. 

§ 239. The maximum height of the tide is not however 
when the moon is in the meridian, but about three hours 
afterward, In consequence of the impulse given to the 



166 ELEMENTS OP ASTRONOMY. 

waters, which indeed is only gradually diminished, they 
continue to flow, and thus accumulate for some time. In 
like manner the highest spring tides do not occur at the 
instant of the sun and moon's acting on the waters in con- 
junction, but sometime afterward. 

The rising tide is called flood, and the falling, ebb 
tide. The average height of tide for the whole globe is 
about 2J feet. If the earth were covered uniformly with 
a stratum of water the difference between the two diame- 
ters of the oval would be five feet. Local causes some- 
times give the tide a height of 70 or even 120 feet. The 
Atlantic on the shores of France and also of North Amer- 
ica pours into the rocky shore with prodigious violence, 
while the gentle tides of the Pacific are scarcely percepti- 
ble. The winds have great influence on the height of the 
tides according as they conspire with or oppose them. But 
the actual effect of the wind in exciting the waves of the 
ocean probably extends very little below the surface. 
Even in the most violent storms the water is probably calm 
at a depth of ninety to a hundred feet. Currents and 
headlands modify the force and the time of the tide for 
each place, and in land-locked seas like the Mediterranean 
and Baltic there are no tides. If the globe were covered 
with islands and continents, leaving no large open sea, there 
would be no regular tide. The variations in the sun's dis- 
tance are too slight to influence the tides much, but those 
in the moon's distance have considerable effect. When the 
moon is near at the time of the equinoxes the very highest 
tides are produced. 

§ 240. The equilibrium of the ocean can never be 
destroyed by the causes which influence the tides ; they 
are sufficient to keep the ocean in perpetual agitation, but 
can never detach it or in any great degree withdraw it 
from the earth. The sun's influence on the ocean is only 
swziwum of gravity at the earth's surface, and the action 
of the moon is little more than twice as much. Had the 
action of the sun and moon been equal there would have 
been no neap tides. In ports where the tides arrive by 
two channels of lengths corresponding to half an interval, 
there is neither high nor low water, on account of the in- 
terference of the two tidal waves. 



ELEMENTS OF ASTRONOMY. 167 

While our place on the earth is at the level of the ocean 
of waters it is at the bottom of the aerial ocean. This rests 
on the waters and fills up the valleys as the sea fills in its 
bed, and probably presents to observers in other planets 
the level, swelling, outline surface which the sea presents 
to us. In it man and animals move, by this they support 
life. All that meets their sight and hearing passes through 
this medium. 

The atmosphere is an elastic fluid of great rarity, and 
vastly more compressible than water. Water being but 
slightly compressible is of very nearly the same density at 
all depths. Air being exceedingly compressible, the lower 
layers are pressed down by the weight of those above, and 
the density is much greater in the lower than in the upper 
regions. A column of air reaching down to the centre of 
the earth and compressed between walls, a few miles be- 
neath the surface of the earth would be dense as gold. 
The lowest part of a similar water-column would have three 
million times the density of common water, and 119 times 
that of most marble. The pressure of the air on every 
square inch is fifteen pounds ; so that the whole globe sus- 
tains a weight of 14,449,000,000 hundreds of millions of 
pounds. 

The atmosphere contains a definite amount of air, as the 
ocean does of water. This is held to the surface of the 
earth and the sea by gravity. Its elasticity contends 
against gravity, and constantly endeavors to enlarge its 
volume and thus remove it farther from the surface of the 
earth. The lower layer is pressed on by all the rest ; its 
elasticity is therefore less available, and it is more dense. 
The next layer above has less pressure from above, and is 
less dense. As the heights above the sea increase in arith- 
metical progression, the density of the atmospheric column 
diminishes in geometrical progression, consequently at a 
much more rapid rate. 

§ 241. By calculation founded on this proportion we 
learn that the atmosphere extends to about forty-five miles 
above the surface of the earth, a distance no greater com- 
pared with the size of the earth than the skin of a peach 
compared with the fruit within. This may not be the 



168 ELEMENTS OF ASTRONOMY. 

exact height of the atmosphere, but it extends at least as 
far as this, because at this height it reflects light, and it is 
evident that there must be some limit where elasticity and 
gravity balance one another. The height of that portion 
of the atmosphere which is sufficiently dense to reflect light 
may also be found rudely from the length of time twilight 
lasts. 

We may learn from the barometer the proportion in which 
the atmosphere is distributed. This indicates, at 1,000 
feet above the level of the sea, that we have left below us 
one thirteenth of the mass of the atmosphere. At 10,600 
feet of perpendicular elevation, (which is rather less than 
that of the summit of Etna,) we have ascended through 
about one third. At 18,000 feet, (which is nearly that of 
Cotopaxi,) we have beneath us one half the atmosphere in 
weight. At an altitude not exceeding the hundredth part 
of the earth's radius the thinness of the air would be so 
extreme that neither combustion nor animal life could be 
maintained. The atmosphere, however, only extends to 
about the eighteenth part of the distance of a radius. Its 
mass is not more than ruuTriva-uTT part of that of the globe. 
The atmosphere is spread over the earth in concentric 
layers without any regard to the inequalities of the surface 
on which it rests. The air in deep mines is the most com- 
pressed, but the communication through this mobile fluid is 
so rapid that storms influence the barometer there as surely 
as on the surface of the earth. At a mile above the level 
of the sea the air all over the earth has the same density 
whether it rests on air or on a mountain. It if were not so 
the barometer would be no test of elevation. 

§ 242. The shape which this atmosphere assumes is 
undoubtedly that of a spheroid, probably more flattened 
than that of the earth, because being more distant from 
the centre centrifugal force bears a larger proportion to 
gravity than it does at the surface of the earth. If the 
earth rotated more rapidly the ocean and the atmosphere 
would be floated from thesurface. It is only the larger 
globes, such as Jupiter and Saturn, which by their greater 
mass can counterbalance the centrifugal force generated by 
their swift rotation. 



ELEMENTS OF ASTRONOMY. 169 

Had this atmosphere been differently composed, man 
could not have retained his present physical constitution. 
The gases which compose the air are adapted to the lungs 
of its inhabitant ; neither gas can be increased without in- 
juring him. The supply of air also is precisely what the 
lungs require. The diver who fills his lungs with air com- 
pressed beneath the water, finds his lungs irritated, while 
the extreme pressure on his ears, his head, and the whole 
surface of his body is extremely painful. No less is the 
suffering experienced by those who ascend lofty mountains. 
The air in their bodies being less rarified than that around 
them causes the most unpleasant sensations in the head and 
ears. The air which they breathe is too rare to satisfy 
their lungs. Fishes cannot exist in ponds on lofty moun- 
tains, for the air in the water has not sufficient oxygen. 
For the same reason it is difficult to kindle or maintain 
a fire. Even sound is feebly transmitted by so thin a 
medium. 

§ 243. In the atmosphere are diffused watery vapor 
and many gases. Electricity, magnetism, light and heat 
are busy there, working changes in organic and inorganic 
nature. The robe of green with which the earth is made 
ready for man and animals has existed in the atmosphere 
in a gaseous state. In this delicate form animals and plants 
imbibe and condense them within their frames to all those 
several forms which the Infinite Disposer has appointed to 
matter. 

But the atmosphere is not only useful, it is the abode of 
all which is most beautiful. We scarcely know which to 
admire most, the pure transparent depths which almost 
open to us another world, or the many forms of loveliness 
assumed by those sky-dwellers the clouds. 

The sun heating the water and the damp earth causes 
the moisture to take a vaporous form and consequently to 
rise. Thus the earth is surrounded by two atmospheres, 
one of air, one of watery vapor, either of which might exist 
independent of the other. This vapor is invisible as long 
as the air which it permeates is sufficiently heated to allow 
it to remain a vapor. When, by ascending or by meeting 

15 



170 ELEMENTS OF ASTRONOMY. 

a cold stratum of air, it is chilled, it ceases to be a vapor, 
it becomes a cloud. 

§ 244. Of the precise way in which clouds are formed 
we are ignorant. Those who have observed them on moun- 
tains say that they consist of minute vesicles so light as to 
float in the air. Perhaps electricity is concerned in their 
formation, and still more in their precipitation to earth in 
the form of hail, rain and snow. 

Most clouds are less than four miles above the sea level, 
and many are far below this, and of course below mountain 
tops. 

Probably clouds never exist at a height greater than 
ten miles, at which height the density of the air is about 
an eighth part of what it is at the level of the sea. Vapor 
rising from the earth is condensed before it reaches this 
height ; for the air becomes rapidly colder as we ascend. 
Air being no conductor of heat is only warmed by contact 
with the heated earth or sea. The heated layer expands 
and consequently rises, and the one which takes its place 
is heated and again rises. The rays of the sun pass 
through it without imparting heat as light passes through 
transparent substances. As the heating of successive 
layers is a slow process, the air is not warmed to any great 
distance above the general surface ; and this is one cause 
of the cold of mountain tops. 

§ 245. The atmosphere like the sea is never tranquil. 
Its extreme lightness, and its expansion by heat and con- 
sequent change of weight, prevent its ever remaining in a 
state of rest. This state of agitation not only equalizes the 
temperature and the moisture, but conduces to the health 
and vigor of vegetable and animal life. It is only when we 
feel the breeze that we realize how great a restorer con- 
stantly envelopes us. Waves and tides on a larger scale 
than those of ocean may and probably do exist at the sur- 
face of our atmosphere ; indeed recent experiments have 
proved this and have even shown a difference between the 
influence of the moon when in apogee and in perigee. 
There must also be an aerial as well as an oceanic equato- 
rial current. As the friction of the particles of water on 
each other prevents their rotating with as much velocity as 



ELEMENTS OF ASTRONOMY. 171 

the mass of the earth, and thus creates a westward ten- 
dency or current particularly in the parts of the earth's 
surface which have the most rapid motion, thus also, but in 
a much less degree does the friction of the particles of air 
cause them to lag behind, and to form a current from east 
to west. Within the atmosphere we find steady currents 
and. unsteady motions, raised by the unequal heating of the 
earth's surface. To both of these we give the name of 
winds. These winds are cold or warm, moist or dry, ac- 
cording to the temperature of the country or ocean from 
which they blow. The most important currents in our 
atmosphere are the trade winds, whose regular return ena- 
bles our navigators to travel with speed and safety over the 
central portion of the globe. They prevail within the lati- 
tudes of 30° N. and 30° S. 

As the zone which extends 23° each side of the equator 
is more nearly in the sun's plane than the other parts of 
the earth, this zone is more heated than the rest. The 
air above it is also heated, expands and rises. Air being 
elastic presses in on all sides to fill up the vacuum. The 
cold and heavy air from the poles rushes towards the 
tropics, and the heated air, in an upper current, flows to- 
ward the poles, cooling gradually as it passes. We might 
therefore expect on the earth's surface a steady south wind 
in the northern hemisphere, and in the southern a steady 
north wind. And this would be the case but for the greater 
velocity of the equatorial portions of the atmosphere. 

§ 246. The air appears to be at rest because it par- 
takes of the velocity of motion of that part of the earth's 
surface on which it rests. Each portion therefore has a 
velocity of rotation proportioned to that of the circle 
of latitude to which it corresponds. That near the poles has 
a very small velocity. Therefore as it rushes toward the 
equator its speed is insufficient to keep up with the rotation 
of the earth and the superincumbent air. Thus the cur- 
rents which otherwise would be north and south, lag, and 
hang back from east to west in the direction opposite to the 
earth's rotation, and become north-east and south-east winds. 
As they approach the equator, the earth, by attraction and 
by its friction, communicates to them greater velocity of 



172 ELEMENTS OF ASTRONOMY. 

rotation, so that near the equator they become more nearly 
north and south. Here these opposite currents meeting 
destroy one another, one or other prevailing locally owing to 
the distribution of land and water near the equator. As 
the earth is of nearly the same size for a degree or more 
each side of the equator, we should expect these currents 
to be much weakened before they reach the equator. 
Accordingly here we find a region comparatively calm and 
free from any steady easterly wind. Ships often are be- 
calmed and wait for weeks to cross the line. 

As the heated equatorial air flows over towards the poles, 
it retains a greater velocity than that of the parts over 
which it passes. Hence it gains on the earth in an east- 
erly direction. As it gradually sinks to the surface it 
causes a south-west wind in the northern hemisphere, and 
a north-west one south of the equator. This is the origin 
of the westerly winds so prevalent on the Atlantic as 
usually to make the passage from Europe to America 
shorter than that in the contrary direction. 

If a portion of the air moving slowly were suddenly 
transferred to that which has a more rapid motion, or the 
reverse, a most violent shock or hurricane would be the 
result. Possibly this may be one of the causes of hurri- 
canes. 

§ 247. Beside the trade winds there are other steady 
local winds, evidently owing to the position of the sun. 
Some of these, as the monsoons, are caused by the alter- 
nate heating of Asia and Africa. A north-easterly wind, 
probably a portion of the northern current moving towards 
the equator prevails in summer all over Europe. 

Other winds arise from purely local circumstances. 
Thus the east winds so prevalent on the eastern coast of 
America in the spring arise from the excess of the heat 
of the continent over that of the sea constantly cooled by 
icebergs and polar currents. The west and north-west 
winds which prevail at other seasons are attributed to the 
gulf stream, which by heating the air above it draws 
down cold air from the neighboring parts. 

Wherever, as on a desert or a rocky plain, the nature of 
the soil is such as to become more heated than the sur- 



ELEMENTS OF ASTRONOMY. 173 

rounding surface, the air is heated above it, and cold air 
rushes in all round. Columns of air once set in motion 
rotate rapidly, bearing all things along with them. Their 
path may be traced by the destruction they leave behind, 
trees and houses lying so as to show that the rotation not 
the onward motion of the column has done the mischief. 
These tornadoes are mentioned here because something like 
them is observed in the sun's atmosphere. Tornadoes and 
hurricanes are more common in the tropics, as we might 
expect. The air having here so rapid a proper motion, a 
greater shock would be caused by any obstacle thrown in its 
way. We do not know enough of the arctic regions to pro- 
nounce with certainty, but they seem to be the regions of 
silence and comparative quiet. The temperate zones have 
been called the battle-ground of the winds. Winds of all 
directions certainly prevail here, but if their battles are 
numerous, they are at least not very severe, mere skirmishes 
compared with the hurricanes of the tropics. 

The velocity of wind varies from mere nothing to a hun- 
dred miles an hour ; and when moving its fastest it has a 
force of forty-nine pounds on the square foot. 

The air at the equator rotates at the rate of 1,000 miles 
an hour, but as we partake its motion we are perfectly in- 
sensible to it. 

§ 248. Since we live at the bottom of the atmosphere 
and are enveloped in it, it is evident that whatever reaches 
us from other bodies may undergo changes Avhich we have 
no means of discovering. The heat which the sun sends 
us may be divested of some of its most lively properties. 
Light, whether from the sun or the stars, may have lost 
some of its constituent parts, or may have altered in color. 
Perhaps but for this blessed canopy the excess of light and 
heat we should receive from the sun would be unendurable. 
Perhaps by some modification of the absorbing and retain- 
ing powers of their atmospheres the remotest and the 
nearest planets, nay even the sun himself, may become in- 
habitable. Air and water are among the most transparent 
bodies which we know, yet these when interposed in suf- 
ficient quantity absorb great quantities of light. On the 

15* 



174 ELEMENTS OF ASTRONOMY. 

summits of the highest mountains, where light passes 
through a much less extent of air than on a plain, a greater 
multitude of stars are visible, and through great depths of 
water objects become almost invisible. 

Since air though rare consists of material atoms we can- 
not be surprised at its effect on light and heat. It has 
weight, and apparently a blue color, and to this the hue of 
distant hills is owing. This color of the air must affect 
the color of all light transmitted through it. We therefore 
do not any more see the sun of its true color than the diver 
in the depths of the sea, who beholds it at noon-day of a 
red color. To him it has undergone two changes in pass- 
ing through two media ; to us, as far as we know, it has 
undergone but one. Although it has been conjectured 
that the dark lines and spaces in the solar spectrum may 
be left by rays of light quenched at his surface. Perhaps 
in the atmospheres of the other planets blue rays may be 
absorbed and red or yellow transmitted, and thus another 
source of variety may be introduced into the system. 

§ 249. Of the amount of all these changes and losses 
we must continue ignorant. We can never know the quan- 
tity of heat or light sent out by the sun, nor the change of 
color light suffers before it reaches us. But we know from 
experiments on transparent media some of the laws of 
these changes, and we can estimate the quantities received 
from him at different altitudes, or at the same time by two 
observers with quantities of air interposed, as at the summit 
and base of the same mountain, and can ascertain whether 
these estimates agree with the laws we have assigned to 
transparent media. 

Two laws of absorption are well established ; first, that 
light is lost in proportion to the obliquity of the incidence ; 
second, that the loss is proportioned to the density of the me- 
dium and the amount of it traversed. The first law may be 
verified in the nearest pond ; vertical rays have more pen- 
etrating power than those which fall obliquely ; they seem 
to meet with resistance proportioned to the obliquity of 
their incidence, and more are therefore quenched or ab- 
sorbed on their passage. If light is a motion of an ether 



ELEMENTS OF ASTRONOMY. 175 

which penetrates all bodies, we can easily conceive that this 
motion should be brought to rest more speedily by an ob- 
lique than by a direct passage through a medium partially 
transparent. Now the rays of the vertical sun enter the 
atmosphere at right angles to its surface, and they enter 
more and more obliquely as the sun approaches the hori- 
zon. From this cause then not so many rays of the de- 
clining as of the noon-day sun penetrate our atmosphere. 

§ 250. Let us consider in what direction the greatest 
quantity of air would be interposed between us and a 
heavenly body. If we draw a globe surrounded by an at- 
mosphere, and from our position on it draw several lines, 
one toward the zenith, one at an angle of 45°, and one 
horizontal, we shall find that the last passes through the 
lower and denser layers for a much greater distance than 
either of the others. The vertical line passes through the 
shortest extent of atmosphere ; and each line passes through 
the air and through the dense layers for a greater distance, 
in proportion as it approaches the horizon. These lower 
layers are not only more dense in themselves, but they are 
usually loaded with vapors and often with clouds. The 
absorbing power of vapor is seen in the clear crimson and 
orange light diffused by the setting sun. The vapors then 
suspended in the air absorb all the other rays, allowing 
passage to these alone. The whole amount of light and 
heat absorbed by these vapors must be very great since it 
enables us to bear the sight of the sun. Hence less light 
is received toward the horizon than near the zenith because 
there is actually a larger body of air and vapors to be pen- 
etrated near the horizon. 

§ 251. As the heat-giving rays apparently proceed 
from the same cause and are propagated in the same way 
as the light-giving rays, and as observation shows the same 
results, we may conclude that they are subject to the same 
laws, and we may state the general law, that the absorp- 
tion of heat and light by the atmosphere increases with the 
obliquity of the incidence. 

Of ten thousand rays falling on its surface, 8,123 arrive 
at a given point of the earth if they fall perpendiculaly ; 
7,024 arrive, if the angle of direction be 50° ; 2,831 if it 



176 ELEMENTS OF ASTRONOMY. 

be 70° ; and only five rajs will arrive through a horizontal 
stratum. Between one quarter and one fifth of the sun's 
light is lost in passing through a vertical plane. In passing 
through a horizontal stratum the light is diminished 1,800 
times, and this enables us to look at the setting sun without 
being dazzled. A haze increases the loss to one third. 

On the summits of high mountains, where but little air 
intervenes between the observer and the. heavens, a multi- 
tude of small stars may be seen which are invisible from the 
plain. Within 10° of the horizon small stars become in- 
visible, because the light from them cannot penetrate so 
dense a medium. 

§ 252. We may form some idea of the quantity of 
light and heat sent from the sun, by considering that they 
are transmitted through space in all directions, and inces- 
santly ; that comparatively few of them fall on any globe 
or object that is known to us, and that if the sun were sur- 
rounded, at the distance of Uranus, by a vast hollow globe, 
it could brighten the walls of that with as much ease as it 
now gilds the few planets which wander in depths of space. 
When we reflect on the vast quantity of heat and light re- 
ceived by our earth, which is but a pin's point in the 
heavens, and on the brightness which the planets and the 
moon owe to the rays which they receive, we can scarcely 
conceive of the immense number of rays continually ra- 
diated and lost in space. 

Various attempts have been made to estimate the light 
which we receive from the sun. Its direct light has been 
estimated as equal to 5,563 wax candles of moderate size 
placed at the distance of one foot from the object ; that of 
the moon is only equal to the light of one candle at the 
distance of twelve feet. Consequently the light of the sun 
is more than 300,000 times as great as that of the moon, 
and that of the moon is too small to afford heat. It would 
require 90,000 moons, enough to fill the whole of our visi- 
ble sky, to afford us light equal to that we have in a cloudy 
day when the sun does not shine out. All the light we 
have in a cloudy day is by reflection from the clouds, and 
the light of the moon is only a reflection of that of the 
sun ; therefore it would take 90,000 of them to give us the 



ELEMENTS OF ASTKONOMY. 177 

same light. Compared with Sirius, one of the nearest and 
largest of the fixed stars, the sun's light is twenty millions 
of millions of times as great as that of Sirius to us. But 
Sirius placed where the sun is would appear 3.7 times as 
large as the sun, and would send out 13.8 times as much 
light. 

§ 253. We will now consider whether the rays which 
enter our atmosphere are bent from their true path. This 
is a more important inquiry, in an astronomical point of 
view, than their diminution. If rays are bent, we can 
never see any heavenly body in its true place, unless it is 
in the zenith. And since to us, in latitude 42°, the sun, 
moon and planets, are never within 12° of the zenith, it 
follow that we never see any member of our solar system 
in its true place, if refraction actually takes place. 

As we live at the bottom of the denser medium through 
which rays pass, it is impossible for us to see their refrac- 
tion as we can when a ray or a stick passes from air into 
water. We can only find out by experiments with trans- 
parent media what are the laws of refraction, and then 
ascertain whether the phenomena observed in the heavens 
agree with those laws. 

§ 254. We find on earth that whenever a ray passes 
obliquely from a thin to a denser medium it is bent towards 
a perpendicular to that point of the surface which it touches. 
It is more bent in proportion to the density of the medium 
it enters, and also in proportion to the obliquity of inci- 
dence. Refractron takes place only at surfaces. At what- 
ever angle a ray enters a medium, at the same angle it 
proceeds through that medium ; therefore refraction does 
not, like absorption, increase with the thickness of the 
layer or the extent traversed. 

A ray falling perpendicularly on a medium is not re- 
fracted, but passes on in its original direction. For ob- 
lique rays the sines of the angles of incidence and refrac- 
tion, (the angles which the ray before and after meeting 
the surface makes with a perpendicular to that surface at 
the point where the ray meets it,) bear always, whatever 
be the amount of the angle of incidence, the same propor- 
tion to each other so long as the medium is the same. 



178 ELEMENTS OF ASTRONOMY. 

If the ray passes through several media bounded by 
parallel plane surfaces, the effect produced by refraction is 
the same in amount as if it had originally fallen on the last 
of these media at the same angle as that at which it fell on 
the first. 

Let us see if in our atmosphere we find the conditions 
of refraction ; and also whether we observe its consequences 
in the altered position of the heavenly bodies. 

§ 235. Though the outer parts of our atmosphere have 
been described as exceedingly thin, the surrounding ether 
is much more rare, or it would affect the motions of the 
planets. Light passes from the ether to the atmosphere, 
from a thin to a denser medium ; therefore it is refracted, 
except when it falls perpendicularly ; therefore we see all 
heavenly bodies, except those in the zenith, higher than in 
their true place. For as we have no knowledge how much 
rays may have been bent since they left bodies, we refer 
all bodies to that direction from which light meets our 
eyes. 

If the air were of the same density throughout, rays 
would be bent once at the surface of the atmosphere, and 
would then reach the earth without again changing their 
angle. But as the strata of the air change continually in 
density as they lie nearer the earth, as they are infinite in 
number and infinitely thin, the refracted ray describes a 
curved path. And since in similar media refraction is 
great in proportion to the density of the compared layers, 
and the density of the lower layers increases so rapidly, this 
curve becomes rapidly steeper as it approaches the earth. 
But we at the bottom of the aerial ocean know nothing of 
this curvature of the ray ; we refer the object to the end 
of a tangent to the curve at the place where it enters our 
eye, and of course see it more elevated above our horizon 
than it really is. If the air were of equal density, the re- 
fracted ray would be the straight line joining the ends of 
its present curve. 

§ 258. The amount of refraction is slightly less on 
the convex surface of the earth than it would be on a plane 
surface, for the perpendiculars drawn from each point of 
the surface of the atmosphere are no longer parallel to the 
zenith line, and thus the angle of refraction diminishes. 



ELEMENTS OE ASTRONOMY. 179 

Since refraction increases with the obliquity of the inci- 
dent rays, it must increase very rapidly near the horizon. 
Its amount is also much increased by the commotions of the 
atmosphere and by the moisture floating in it. Strata of 
different density laid one upon another also have great re- 
fractive power. Since the fluctuations of heat, winds and 
clouds, never extend more than ten miles, and usually take 
place at a much less height, it follows that refraction is 
much greater in these lower regions. And as horizontal 
rays pass through many more of these clouds and vapors, 
it is not surprizing that refraction increases rapidly as we 
approach the horizon. 

The amount of refraction may be found by observing the 
greatest and least altitudes of some circumpolar star which 
passes at or near the zenith. Then knowing the latitude 
of the place, the distance of the star from the pole at each 
observation will also be known ; as the star is not influenced 
by refraction in the zenith, the differences of these dis- 
tances will be the refraction at the least altitude. The in- 
fluence of refraction in every part of its course, except 
when in or very near the zenith, is shown by the star's de- 
scribing a flattened curve instead of a circle. The varia- 
tion is however too slight to be perceived by the naked eye ; 
it differs at each instant. 

§ 257. The law which refraction follows has been de- 
termined, but so many accidental causes modify its amount 
that observation is found to be a better guide than theory 
in constructing the tables which are indispensable to cor- 
rect every altitude of a heavenly body. In the zenith the 
refraction is nothing ; at 45° it is about V, a quantity 
scarcely perceptible by the naked eye ; in the visible hori- 
zon it is 33', which is rather more than the greatest ap- 
parent diameter of the sun and of the moon. Thus we 
see the sun's whole disc at rising and setting when it is 
actually below the horizon. And in some parts of the year 
it rises five minutes earlier in the morning, and sets five 
minutes later in the evening, than if there were no refrac- 
tion. The mean time added from this cause is three 
minutes. 

Since refraction takes place in our atmosphere, it of 



180 ELEMENTS OF ASTRONOMY. 

course affects all the heavenly bodies however distant. 
In 1750 a singular consequence of refraction was observed 
at Paris. The moon suffered eclipse in the shadow of the 
earth, although both sun and moon were above the horizon, 
one in the west, the other in the east. The sun was in 
reality a little above the horizon, the moon a little below 
this plane, but it was raised and made visible by refraction. 
If the sun had been a little below and the moon a little 
above, the same phenomenon might have taken place. 

§ 258. Refraction was first suspected from the dif- 
ferent apparent heights of the same star at different sea- 
sons, and from the different distances of the same star from 
the polar star, according as it was more or less near the 
zenith. This was at first attributed to accidental vapors, 
and it was not suspected that there was always some re- 
fraction. It is now known that refraction takes place every 
where except at the zenith, although the amount of it at 
the same height is varied by many circumstances. By 
measuring the horizontal refraction of the upper limb of the 
sun when it first appeared in the horizon, and then its lower 
limb, it has been found that even while the sun was rising, 
refraction had diminished 25". 

Even terrestrial objects are influenced by refraction. 
The altitude of a hill is sensibly greater on a cloudy, dull 
day, when the air is thick and heavy, than on a clear day. 
It is also greater before sunrise than at the noon of a bright 
day. Not only the barometer but the thermometer indi- 
cates by its changes a change of refraction — increased cold 
being accompanied by increased refraction. From the in- 
crease of refraction in night and in winter, it has been sup- 
posed that refractions are proportionately greater toward 
the poles, and serve to shorten a little the gloomy polar 
night. 

§ 259. Refraction has a very singular effect on the 
form and proportions of bodies in the horizon. It has the 
same upon the form of bodies under water seen from the 
air. A ring plunged in water looks flattened, because 
the lower part is more uplifted than the upper. Thus the 
sun, which always appears round when in the zenith or at 
an altitude of many degrees, as it approaches the horizon 



ELEMENTS OF ASTRONOMY. 181 

becomes flattened and apparently oval, because the refrac- 
tion of the lower limb being greater than that of the upper, 
the vertical diameter is diminished. The horizontal diam- 
eter is very slightly diminished, because the vertical lines 
in which refraction takes place are vertical circles. The 
convergence is however extremely small. For suppose the 
diameter of the sun to be 32', and the lower limb to touch 
the horizon, then mean refraction at that limb would be 
33', but the altitude of the upper limb being 32', its re- 
fraction is only 28' 6". The difference between these re- 
fractions is 4' 54", the quantity by which the vertical 
diameter appears shorter than that parallel to the horizon. 
When a body is not very near the horizon refraction dimin- 
ishes very nearly uniformly. 

Although the lower limb of the sun and moon appear 
expanded when in the horizon, the whole disc is smaller. 
Their splendor is diminished, and the cause of this is so 
frequently increased distance, that we are apt to suppose it 
the only cause, and to refer dim and indistinct objects to a 
great distance. Painters expect this reasoning from us ; 
they use indistinct outlines and faint colors for distant ob- 
jects, bold outlines and strong colors for near objects. 

After we have referred a dim object to a greater dis- 
tance than the true one, by another process of reason we 
judge it to be larger than it really is, because at so great 
a distance it appears of a given size. 

We refer all bodies near the horizon to a greater distance 
than we do when they are overhead, because they are all 
thus dimmed, and we extend this illusion to the form of the 
arch itself, and believe that we see it more extended near 
the horizon than overhead. Whereas in fact we see verti- 
cally to a greater distance than we do horizontally, for a 
small star which is visible near the zenith, is invisible near 
the horizon. So that our vision does not extend to quite a 
concave hemisphere, it is narrowed near the horizon. The 
arch does not at all times appear of the same elevation. 
It is higher in clear than in dull weather. 

§ 260. Perhaps one reason that a horizontal section 
of the arch appears larger is that we have more means of 
measuring it. As we look toward the horizon we see stars 
16 



182 ELEMENTS OF ASTRONOMY. 

separated from one another by space, extending in a long 
series, or we see an alternation of clouds and sky, and be- 
neath, a landscape which we know extends to a great dis- 
tance. But if we look upwards we have no line of stars, 
no marks to inform us whether we are gazing miles, or hun- 
dreds, or millions of miles, into the heavens. 

In the same way our judgment of the size of the moon 
in the horizon is affected by the vicinity of objects so much 
smaller as terrestrial objects must be. 

Actual measurement proves that when they are in the 
horizon the sun subtends the same, and the moon a much 
smaller angle, than when at a greater altitude. Any one 
may satisfy himself of this by rolling a piece of paper into 
the form of a tube, making the opening the size of the 
moon when in the horizon. Tie a thread round it to keep 
it of the same size, and when the moon comes on the 
meridian and appears much smaller to the eye, look at her 
through the tube, and she will appear larger than at her 
rising. She will appear larger because she is nearer to 
the observer ; of course her actual size cannot vary, for 
when she is in the zenith of one observer, she is in the 
horizon of another. Suppose her in the zenith of a person 
90° distant from us, and in our horizon. It is evident her 
distance from us is greater than her distance from the 
other observer. If her distance is greater she subtends a 
smaller angle and therefore appears smaller. 

Another phenomenon of a somewhat similar nature, is 
the apparent enlargement of the bright part of the moon 
when both the bright and dark parts are visible. The une- 
qual impressions made upon the retina of the eye Jby the 
bright and the feebly illuminated regions give rise to this 
illusion. 

§ 261. Refraction likewise occasions twinkling or undu- 
lations in the light of the fixed stars. The atmosphere be- 
ing very easily expanded by heat and condensed by cold, 
is always more or less agitated. The layers of molecules 
which compose it experience momentary condensations and 
dilatations, which cause the direction of the luminous rays 
to vary incessantly by the difference of refraction which 
they occasion. Yapors and layers of air of different densi- 



ELEMENTS OF ASTRONOMY. 183 

ties are also drifted rapidly along, so that the' refractive 
power of the medium varies continually. These effects are 
almost always perceptible in our country, because the air 
here is seldom serene ; they are less so in countries where 
the sky is more pure. They occur particularly when the 
weather is changing. For this reason the fixed stars, whose 
apparent diameter is very small, appear to us agitated by 
a sort of trembling. This happens especially just before 
rain when it succeeds a long dryness. The twinkling of 
the stars is then so remarkable that it becomes a sign for 
sailors. 

At such times if a star is observed with a delicate tele- 
scope, when the star is placed under the thread, it will 
oscillate so as to appear successively on each side the 
thread. These motions succeed one another with such 
rapidity that the visible diameter of the star appears to 
exceed the thickness of the thread. These motions are 
sometimes so great that it is impossible to observe. 

Very marked agitations, produced by the same cause, 
may be observed in the shadows of towers, and in the 
image of the sun projected on the ground by an opening 
made in the dome of a lofty building. 

The rapid motion of the air caused by dilatation may be 
seen above stoves, or above fields and the roofs of houses 
when much heated after a long drought. Any object seen 
through this rapidly changing medium appears distorted 
and trembling. 

We are apt to suppose that the fixed stars look larger 
than they really do, owing to the false glare occasioned by 
this trembling. An ordinary telescope magnifies them still 
more than the changeable air. The imperfections of the 
glass give the stars spurious discs. But on observing these 
same stars with telescopes of a much higher power they 
appear as mere points. 

The planets twinkle much less than the fixed stars ; 
their discs are so much larger that they cannot be dis- 
placed totally. They experience on the edges little undu- 
lations, while the stars which seem but brilliant points are 
continually displaced. This displacement produces twink- 
ling. 



184 ELEMENTS OF ASTRONOMY. 

§ 262. Refraction is not the only change light expe- 
riences in our atmosphere. Reflection does not influence 
the distant luminaries, but acts at all hours and on every 
body within our atmosphere. 

It is not easy to separate the effects of refraction from 
those of reflection. Twilight, for instance, is the effect of 
reflection following refraction ; reflection prolongs it more 
than refraction, but to refraction and absorption we owe 
the infinite variety of the morning and evening sky. 
Without their influence there would be a sudden transition 
from splendor to darkness, from day to perfect night. As 
refraction brings up the sun's disc when actually below the 
horizon, so it afterward brings his rays up higher, and 
makes them visible longer than they would otherwise be. 

Reflection causes the rays from a body below the horizon 
after rising above the horizon and striking against the va- 
pors and clouds, and perhaps the atoms of air, to be sent 
downward to the earth. If the sky is clear these rays are 
reflected to us of a pure yellow light. If it is loaded with 
clouds and vapors at different heights, different colored 
rays struggle through these, and we have a sky varying in 
color every instant as the rays strike the clouds more or 
less obliquely. The red rays have most momentum and 
therefore pass through a misty sky where no others would. 
The other colored rays are absorbed. The sun's rays rise 
sufficiently high in our atmosphere to be reflected until he 
is 18° below the horizon. 

§ 263. The usual duration of twilight in the temperate 
zone is an hour and a fifth long. Duration of twilight is 
increased even more than we should expect on high moun- 
tains. De Saussure passed several nights on the high 
Alps, and saw the whole horizon surrounded with pale but 
distinct light which lasted from sunset to sunrise, although 
the sun must in the middle of the night have been 45° 
below the horizon. This reflection did not come from the 
layers of air where the observers stood, for on such heights 
they are so near that their reflection is very feeble, but 
from the thick and deep mass of air which borders the 
horizon on all sides. 

Analogous phenomena are sometimes seen during an 



ELEMENTS OE ASTRONOMY. 185 

eclipse of the moon. Her disc is not in different eclipses 
always of the same color. It has been supposed that when 
that portion of the sun's atmosphere which is so situated 
as to reflect the sun's rays upon the moon is laden with 
vapor, it gives to the moon the peculiar light and color 
which are sometimes observed. 

§ 284. Reflected light is not seen only in twilight. 
Almost all the light which falls upon our eyes has been 
again and again reflected. The light which comes from a 
bright luminous body is too brilliant to be agreeable ; it is 
painful to the eye. If it fell upon bodies and were only 
once reflected to the eye, we should see on every object a 
round brilliant image of the sun, such as is reflected from 
polished steel and from water. The moon would send us 
only a reduced image of the sun. Every body in the 
direct rays of the sun would be painfully brilliant, and all 
other bodies would be in the deepest darkness ; every room 
into which the sun was not shining would be as dark as in 
the night. But most objects which the sun shines on are 
too rough to reflect his image ; they break his rays into 
innumerable smaller rays, and the greater or less bright- 
ness of these and the angle at which they touch our eyes, 
teach us the form of the body. In very distant bodies we 
lose the difference of shade and the form consequently. 
Thus the sun and the moon from their great distance show 
a flat disc. Besides we see no bodies by the direct light 
of the sun only, but also by an infinity of cross lights, 
which coming to us from all parts of the body show us its 
whole shape. These cross lights on the surface of the 
earth arise partly from reflection from large objects, but 
more from reflection from vapors and small particles float- 
ing in the air, and perhaps also from the particles of air 
themselves. The power of very small particles to reflect 
light is shown by the path of a sunbeam across the room, or 
across a moist atmosphere, when the sun is improperly said 
to draw up water. Particles of dust and vapor, before in- 
visible, become perfectly luminous, that is break and reflect 
the rays in all directions, so that they are themselves seen. 
Undoubtedly much smaller particles have this power, and 
by their unseen action produce the soft generally diffused 
16* 



186 ELEMENTS OF ASTRONOMY. 

light of day. The rays are broken and sent as messengers 
in all directions, crossing and recrossing and wrapping us 
in a perfect web of light, till we almost forget that the 
little disc which our two hands can shut out from our sight 
is the source of it all. 

§ 265. The indirect light which is reflected from the 
sky is often very considerable. In Edinburg it amounts 
perhaps to 30° or 40° of the photometer in summer, and 
10° or 15° in winter. This secondary light is most pow- 
erful when the sky is overspread with thin fleecy clouds. 
It is feeblest in two very different conditions, — either when 
the sun's rays are obstructed by thick clouds, or when the 
atmosphere is quite clear and of a pure azure tint. In 
higher regions, the direct rays of the sun, not being im- 
paired by a long passage through the atmosphere, are more 
vigorous than at the surface of the earth. But the diffuse, 
indirect light of the sky, being reflected from a rarer mass 
of air, unstained by vapors, is proportionately feeble. The 
silvery hue of the sky changes to a dark hue, slightly 
tinged with blue in the day time, and at night serving as 
a transparent black ground for the multitude of stars. As 
this feeble diffused light does not interfere with vision, 
large stars and planets are visible from the shade even in 
the day time. 

Reflection from the ground and other opaque objects 
makes no inconsiderable addition to the amount of light. 
From a sandy beach, the reflected equals one third of the 
incident light. From a wide surface of snow, it amounts 
to five sixths of the direct light ; the numerous facets of 
the bright snowy flakes, which are presented in every pos- 
sible position, detaining only one sixth of the incident rays, 
and scattering the rest in all directions. 

The laws and facts thus far studied concern the globe as 
a whole. Before entering on those which take place in 
portions of its surface, we will see what effect the position 
of an observer has on the appearance of the heavenly 
bodies. 



ELEMENTS OF ASTRONOMY. 187 



CHAPTER XII. 

PHENOMENA WHICH DIFFER IN DIFFERENT PARTS OF 
THE EARTH. 

Day and Night. Circle of Illumination. Twilight- The Seasons. Curve 
traced by the Sun's combined daily and yearly Motions. Portion of the 
Heavens visible in different Latitudes. Length of Day and Night. 
Equinoxes. Effect of Twilight. Amount of Light and Heat received in 
a given place. Equality of the distribution of Heat in the Northern and 
Southern Hemispheres. Difference in their respective Seasons. 

§ 266. Before we enter on those celestial phe- 
nomena which appear different when viewed from dif- 
ferent parts of the earth, we must inquire what effect 
our position on the surface of the earth has on exter- 
nal phenomena. The utmost distance by which two ob- 
servers on the earth's surface can be separated is 8,000 
miles ; and 8,000 miles is an appreciable distance to bodies 
no more distant than the sun, moon and planets. It is 
sufficiently large to be distinguished from them, and causes 
them to change their places as viewed from the earth. 
This displacement of a body from its true place, as seen 
from the earth's centre, is called parallax. Owing to it we 
never see a heavenly body in its true place unless when 
the line joining it and our eye passes through the centre of 
the earth. The displacement is greatest when the body is 
in the horizon of the observer, because the observer is then 
distant from the line joining the body to the centre of the 
earth by the greatest possible amount, the earth's radius. 
It is then called the horizontal parallax. Its amount may 
be observed for the sun, moon and planets ; it is greatest 
for near bodies, less for those more distant, and inappre- 
ciable for the fixed stars. It is less for each body accord- 
ing to its height above the horizon ; it is only in the case 
of a body in the zenith that it becomes nothing, and in 
high latitudes this can never take place with any member 
of the solar system. It always acts in a vertical circle, 
and always depresses the body, thus partially counteract- 
ing the effects of refraction. 



188 ELEMENTS OF ASTRONOMY. 

§ 267. The laws we have hitherto investigated have 
concerned the globe as a whole. We will now study some 
phenomena which are unlike in different parts of the globe, 
the variations of seasons and in the length of days. An 
astronomical day includes twenty-four hours, a natural day 
may be of any length between nothing and six months. 
The various modes of reckoning days and years will be 
mentioned hereafter ; at present we have only to account 
for the length of days at different latitudes, and in the 
same latitudes at different parts of the year. 

To do this we must return to the position of our earth 
and its two motions ; and we must imagine the equatorial 
and the ecliptic to be marked out on the concave sphere 
of the heavens ; and must remember that while the earth 
by its rotation causes- the sun to appear to move from east 
to west in the equator or parallel to it, by its motion of 
revolution it causes the sun to appear to move in the eclip- 
tic from west to east. 

One half of this earth's surface is always illuminated, 
the other half in darkness. This illuminated hemisphere 
has its edges bounded by a great circle called the circle of 
illumination. In the spring and autumnal equinoxes it is 
bounded by a meridian, called the solstitial colure. At 
no other season of the year is it bounded by a meridian. 
The illuminated hemisphere extends 90° in every direction 
from the point to which the sun is at each moment vertical. 
This point is always in the ecliptic. The illuminated 
hemisphere therefore may extend 23° beyond either pole, 
or fall 23° short of it. Since rotation never allows the 
same spot to remain beneath the sun's vertical rays a mo- 
ment, the earth each moment turns up a different hemi- 
sphere to be illuminated. 

§ 268. Let us suppose a concave hemisphere of light 
to be fastened directly between the earth and the sun in 
the plane of the earth's revolution, and to move round in 
the course of the year so as to represent the light falling 
from the sun. Then let us imagine the earth performing 
her rotation in a plane 23° inclined to this hemisphere of 
light, and we shall see that the space for 23° round the 
pole will rotate sometimes entirely in light, sometimes en- 
tirely in darkness. 



ELEMENTS OF ASTRONOMY. 189 

Let us begin at the vernal equinox ; at this time the 
illuminated hemisphere extends to both poles, and just one 
half of the northern and one half of the southern hemi- 
spheres are illumined. The next day at noon, in the same 
place, the centre of illumination lies north of the equator 
a few minutes of a degree, therefore the illumination ex- 
tends a few minutes over the north pole, and falls a few 
minutes short of the south pole. Its edges no longer coin- 
cide with terrestrial meridians ; more of the northern than 
the southern hemisphere of the earth is included in it. 
After another rotation the centre of illumination is farther 
north. And this continues for one quarter of a year. At 
this time the illumination extends 23° beyond the north 
pole ; it has covered the north pole for one quarter of the 
year, and will continue to do so through the next quarter. 
For the next three months the central point advances in 
the ecliptic, always approaching nearer the equator. At 
the autumnal equinox the north pole is left out of the illu- 
mined circle not to enter it for six months. For the next 
six months the illumination creeps slowly over the south 
pole, and then retires from it, till the vernal equinox again 
equalizes the northern and southern hemispheres. In its 
daily revolution the earth turns up 15° of the equator 
every hour. Therefore the illumination extends 15° more 
to the westward every hour. Thus in the course of 
twenty-four hours every meridian has entered the circle of 
illumination, been under the sun, and left the circle of illu- 
mination. 

§ 269. We have said that the sun illumines a hemi- 
sphere, or 90° in every direction from the vertical point. 
It in fact illumines a little more than this, for the rays 
which proceed from the outer portions of the sun, illumine 
a small portion of the otherwise dark hemisphere of the 
earth. For if the shadow cast were cylindrical, one hemi- 
sphere would be illumined, and the rays from the edge of 
the sun's disc would graze the edge of the earth's disc. In 
this case the sun would be precisely the size of the earth. 
And it would make no difference whether the distance be- 
tween them were great or small. 

But let us suppose the sun smaller than the earth. It 



190 ELEMENTS OE ASTRONOMY. 

would then illumine less than half the earth's surface, be- 
cause the rays from the sun's disc, diverging, would meet 
the earth's surface before they reached the large circle 
running at right angles to the sun's direction, which divides 
it into hemispheres. And the ring between this great circle 
and the circle formed by the sun's rays would be in dark- 
ness. In this case if the sun were near, the ring would be 
broader than if he were more distant. 

But since the sun is larger than the earth, the outer 
rays which strike the earth converge, and not only one 
hemisphere but a ring adjacent to it must be illumined. 
If the sun were of the same size he now is and nearer to 
the earth, this illumined ring would be broader than it now 
is. But as he is 95,000,000 of miles distant, its breadth 
is only equal in minutes of a degree to 15', about one half 
the sun's apparent diameter. Owing therefore to the 
superior size of the sun, a ring-shaped surface, 15' in 
breadth encompasses the illumined hemisphere and is added 
to it. 

§ 270. If the earth had no atmosphere this would be 
the only addition to the illuminated hemisphere. But we 
have seen that refraction suffered in the atmosphere makes 
bodies visible 33' before they are above the horizon. 
Therefore refraction extends still further the illumination. 
It adds a fringe of light 33' broad. 

That portion of the earth's surface which lies in twilight 
cannot properly be said to belong to the illumined or to the 
dark hemisphere. Twilight is usually reckoned to last 
until small stars become visible, and this is usually when 
the sun is 18° below the horizon. 

If then we are considering how large a portion of the 
earth's surface is fully illumined, we should say it extended 
90° + (15'+33'=49') from the place to which the sun is 
at each moment vertical. For since the band 15' is act- 
ually illumined refraction acts beyond it. Beyond both of 
these bands is the ring 18° wide which is faintly illumined. 
Thus we have 90°— 18°— 49'=71° 11' for the radius of 
the convex surface which remains in darkness. 

Even here, however, we cannot say with certainty that 
some light is not received from the sun. If so it has un- 



ELEMENTS OF ASTRONOMY. 191 

dergone so many reflections in the atmosphere as to be ex- 
tremely faint. Probably it is only toward jche centre of 
the dark portion that the darkness is as great as if the sun 
were blotted from the heavens. 

§ 271. Let us now consider what points of the earth's 
surface will enter the centre of illumination in the course 
of each day and of the year. Or what is more easily con- 
ceived of, let us consider to what points of the earth's sur- 
face the sun will be vertical, beginning with the vernal 
equinox, on about the 21st of March. Let the earth at 
noon on this day present to the direct rays of the sun that 
point which is common to the planes of her rotation and 
her revolution, and thus the sun will at noon be vertical to 
this point. 

But this point is beneath the sun only the smallest frac- 
tion of time. The incessant rotation of the earth with- 
draws the point from the vertical sun, substituting first one 
and then another and another point, each west of the pre- 
ceding, until in twenty-four hours it has been noon to every 
point in a curve not coinciding precisely with the equator, 
but beginning in it, surrounding the earth, and gradually 
rising, never so much as 24', and seldom nearly so much, 
north of it. The course which the vertical sun marks 
out on the earth the next day is a continuation of this 
curve, almost parallel to the former, but ending a little 
farther north. The whole course of the sun is a spiral, 
made up of these curves, lying one above and slightly in- 
clined to each other, like threads wound skilfully on a ball. 
The threads should not, as we shall explain hereafter, all 
be equally inclined one to another. 

§ 272. For three months this spiral winds gradually 
northwards, the path one day being almost parallel to that 
of the preceding, and almost coinciding with it. At this 
time the north pole is inclined toward the sun, and the 
most northerly part which ever comes beneath the vertical 
sun is now exposed to it. As the earth moves on with her 
pole fixed, the sun will no longer be vertical to this north- 
erly point, but to one nearer and again nearer to the 
equator. For several days the vertical sun describes cir- 
cles nearly parallel to the equator ; the sun appears noon 



192 ELEMENTS OF ASTRONOMY. 

after noon in the same spot — it appears to stand still in the 
heavens. It is called the summer solstice, (from sol and 
stat). The reason of this apparent standing still is that 
the ecliptic and the equator are really more nearly parallel 
in this part than elsewhere, so that the latter degrees 
he ascends differ less than any others. And also that if 
we take the day when the sun described the most northerly 
circle, and compare that with the one preceding and the 
one succeeding, the variation during these three days is 
very much less than the difference in decimation be- 
tween the place of the sun any other three successive 
noons — as when we run up and then make one step down 
a hill. 

The spiral which we have traced for three months con- 
tinues for the next three months to wind gradually descend- 
ing curves round the earth till, in the autumn, it reaches 
the autumnal equinox. It still descends for three months 
more till it reaches the southern solstice, and then ascends 
for three months, having described around the earth 365 
curves, almost parallel to one another. 

§ 273. Thus the apparent path of the sun is caused 
by our two motions. On account of our revolution round 
him, we think we see him describe a great circle round us 
once a year, from west to east. And, since this revolution 
is not in the plane of our rotation, but one half above and 
one half below it, and - 3 i T of it must be described every 
day, the sun, in order to pass through it, is obliged to fall 
nearly one degree short of a circle, and also to move a 
little farther north or south every day. Since our rotation 
is always at right angles to the equator, the sun's daily 
path must be parallel to it, allowing only for his slight 
change in declination ; this change being divided among 
24 hours, and being at most 24', is not perceptible. 

By this simple and beautiful arrangement, the point of 
greatest heat, which if stationary would destroy vegetation 
and animal life, traverses the globe incessantly, returning 
to the same spot only once a year, and then, owing to 
other causes, to a place not precisely the same. While for 
90° in every direction from this spot the earth is warmed 
and illumined. 



ELEMENTS OP ASTRONOMY. 193 

§ 274. Let us suppose one spectator at the pole, 
another at the equator, and a third in some latitude be- 
tween the two ; these are the only important varieties of 
position which can occur. We will inquire how many and 
what stars are seen by each observer at night, and in what 
cases and how the stars seen vary at different seasons of 
the year. We will learn the apparent motions of the stars 
and sun to each j)lace, and the length of time they remain 
visible. 

In all these examinations we shall begin with the posi- 
tion of the globe at the vernal equinox, because at that 
season the planes of rotation and revolution intersect, and 
the sphere is right for all but the inhabitants at the pole. 
As soon as an observer is carried out of the illuminated 
surface, the sun is below his horizon. At that moment, 
were it not for twilight, he would be in darkness. 

§ 275. To an observer on the equator, in the vernal 
equinox, the moment the sun is below the horizon twilight 
begins and lasts till he is 18° below. Were it not for twi- 
light the observer would immediately begin to see the stars, 
and would see a hemisphere of them extending from pole 
to pole and as far as to within one degree of the sun. As 
the earth turns round he would in an hour lose sight on the 
west of all the stars contained from pole to pole within two 
hour circles, and he would gain sight of a new strip of the 
heavens contained on the east within two hour circles and 
reaching from pole to pole. Every hour of the night makes 
a strip of the eastern sky visible, and leaves a strip of the 
western sky out of sight. In the course of the night all 
the stars of the heavens would be seen were it not for 
twilight, for at the dawn of a twelve hours' day an observer 
is 180° from where he was at sunset the night before ; and 
as his horizon in both cases extends over half the celestial 
sphere, he would in the course of the night see all of it. 
Twilight makes a strip, extending from 18° east of the sun 
at night to 18° west of him in the morning, invisible. As 
the earth moves on in her orbit, the sun is referred to dif- 
ferent stars. Different stars are therefore concealed after 
his setting and before his rising. The invisible portion ex- 
tends from pole to pole, and is 36° in breadth at the equi- 
17 



194 ELEMENTS OF ASTRONOMY. 

noctial. Since the earth advances nearly a degree daily 
in her orbit, it will require only thirty-seven or thirty-eight 
days to render this whole strip visible. Every night, how- 
ever, different stars disappear after the sun. Those which 
set some time after him one night follow him immediately 
the next. And those which rise with him one day, the 
next day rise before him, and he is immediately preceded 
by stars before unseen. No two nights in the year have 
the same stars been in the meridian at midnight. Those 
which passed it at twelve o'clock one night pass before that 
the next night. 

§ 276. The observer at the equator is, as we shall 
hereafter show, the only one who, even in the course of a 
year, sees all the stars. His horizon always stretches from 
pole to pole. The stars describe paths at right angles to 
his horizon, and they rise to him always in the same place, 
though not always in the same time. They appear to move 
directly from east to west, and all describe in the heavens 
parallel arcs. 

An observer at the poles would have the same hemi- 
sphere of stars always visible sweeping round his horizon 
in parallel circles. Those near the horizon would appear 
to describe large circles, those near the zenith smaller 
circles. Over head would be the polar star, to him as to 
all other spectators, immovable. Revolution never makes 
more stars visible, but it makes the whole hemisphere in- 
visible during the six months' day while the sun is above 
the equator. A man at the poles then sees only half the 
stars which the inhabitant of the equator sees y and these 
only for one half of the time. 

If the sun were always in the equator he would be 
always in the horizon of an observer at the poles, and thus 
he might live utterly unconscious of the stars and of all 
but the brightest planets. He would see the moon as a 
faint white cloud. 

§ 277. To an observer between the poles and the 
equator the fixed stars have their nightly courses oblique 
to the horizon, and more oblique in proportion as the ob- 
server is nearer to the pole. 

If an observer at the equator has both poles in the hori- 



ELEMENTS OF ASTRONOMY. 195 

zon, an observer ever so little north of the equator loses 
sight of the south pole and sees a little beyond the north 
pole. Let the observer travel 10° north of the equator ; 
then all the, stars within 10° of the south pole will sink 
below his horizon and remain always invisible. All clusters 
within 10° of the north pole will be in his horizon during 
their whole course. They describe circles round the pole, 
being half the time above and half the time below it, and 
are called circumpolar stars. To such an observer 10° 
from the pole is the circle of perpetual apparition. 

If he advances 20° north of the equator, the circle of 
perpetual apparition will extend 20° from the pole. How- 
ever far he advances, the altitude of the elevated pole 
above his horizon always equals his latitude, and all the 
stars between the pole and his horizon will be perpetually 
visible. They will revolve round the pole, keeping their 
relative position and configuration. The Great Bear will 
have his feet always turned from the pole, the Little Bear 
will have his feet toward it. The stars out of the circle 
of perpetual apparition rise obliquely and describe very 
large arcs of circles, more than semi-circumferences. As 
the traveller approaches the pole, they rise with a smaller 
angle to the horizon, and at last the arcs are almost parallel 
to his horizon. Those which rise precisely in the east de- 
scribe a semi-circumference which does not pass through 
his zenith, and remain in sight precisely twelve hours, and 
set precisely in the west. The arc described by these stars 
coincides with the celestial equator. Since the rational 
horizon is a great circle of the sphere, when it does not 
coincide with the equator, it must bisect it and be bisected 
by it. This intersection will take place in the east and 
west points of the horizon, 90° from the north and south 
points of the horizon. 

§ 278. The great circle which passes through these 
.east and west points and through the zenith is called the 
prime vertical. When stars come to the prime vertical 
they are said to be in the east. A body on the same side 
of the equator with the beholder rises between the east and 
north, and comes to the prime vertical after it has risen ; 
a body in the equator rises in the east ; one on the oppo- 



196 ELEMENTS OF ASTRONOMY. 

site side of the equator rises between the east and south, 
and has passed the prime vertical before it rises. A body 
on the same side of the equator with the spectator is longer 
above than below the horizon ; one on the equator is as 
long above as below it ; one on the other side of the equa- 
tor is below longer than above it. 

The nearer the observer is to the equator the greater 
is the number of stars he can see. The circumpolar stars 
are always the same for a given latitude ; their change of 
place alone marks the different hours of the night. But 
all the other stars rise and set, new ones constantly appear 
in the east, describing arcs similar to those of their prede- 
cessors, and disappear toward the west. Since the ob- 
server at the equator in a few nights sees all the stars, and 
one at the poles never sees more than half, observers be- 
tween the equator and the poles see between the whole 
and the half, and more as they approach the equator. 

§ 279. To two observers in the same longitude, but 
differing in latitude, the heavens present different aspects 
at all moments. The stars which are common to both de- 
scribe circles differently inclined to their horizons, and dif- 
ferently divided by them, and attain different altitudes ; 
and some stars are seen by one and not seen by the other. 
To observers situated on the same parallel, of latitude, and 
differing only in longitude, the heavens present the same 
aspects, but at different times. Their visible portions are 
the same ; and the same stars describe circles equally in- 
clined and similarly divided by their horizons, and at- 
tain the same altitudes. In the former case there is, 
in the latter there is not, any thing in the appearance of 
the heavens, watched through a whole diurnal rotation, 
which indicates a difference of place in the observer. The 
only way for an observer in north latitude to increase the 
number of stars visible is to travel toward the south. As 
he moves southward, on the same meridian, most of the 
stars before seen change their places and times of rising 
and the angles they make with his horizon, and they re- 
main visible longer. Those which rise with the equinoc- 
tial, rise in the same time and place as before, but they 
first make oblique and then right angles with his horizon, 



ELEMENTS OE ASTRONOMY. 197 

and when he has passed the equator will appear to describe 
arcs inclined toward the north. The stars round the south 
pole now become perpetually visible ; those within an equal 
distance from the north pole remain invisible. The change 
of stars is more noticed as the horizon advances south, be- 
cause at some seasons of the year and some hours of the 
night very remarkable stars and constellations appear in 
the southern hemisphere. 

§ 280. The apparent path of the sun for each of the 
three observers varies in the same way. If a place is on 
the equator the sun will always rise at right angles to his 
horizon. If a place is at the pole, the sun will move al- 
ways parallel to its horizon. If a place is any where be- 
tween the tropics and the pole, the sun will rise obliquely 
to its horizon, and more obliquely as the place is nearer the 
poles. If it is between the equator and the tropics, it will 
rise twice a year at right angles, and the rest of the year 
obliquely. To a place on the equator at the vernal equi- 
nox, the sun rises precisely in the east, passes through the 
zenith, giving twelve hours of daylight, and sets in the 
west. On the next day, at the same time, it rises five 
sixths of its own diameter further north, and sets after 
twelve hours as much further north of west as it rose north 
of east. The arc described is parallel to the arc described 
the preceding day, but does not pass through the zenith. 
The next day it again rises five sixths of its diameter fur- 
ther north, and at the same hour, for the horizon of the 
man at the equator covers one half the hour circles of the 
globe, describes another arc nearly parallel to the equator, 
and sets further north than the day before. This continues 
for three months, till the sun has risen and set as far north 
as it ever goes, that is to say, 23° 28' north latitude. As 
it approaches the solstice, however, it rises only one sixth 
of its diameter more north. The arc it describes on our 
midsummer day to those in the equator begins 23° north of 
the eastern point, passes 23° north of the zenith, and ends 
23° north of the western point of the horizon. It is the 
lowest arc the sun ever describes to those on the equator ; 
but like all the others it occupies twelve hours. 

17* 



198 ELEMENTS OF ASTRONOMY. 

If we consider the motion of the observer instead of 
the apparent motion of the sun, we should describe these 
changes thus :— In the vernal equinox the inhabitant at 
the equator is carried through the centre of illumination. 
Each day he passes farther and farther south of this centre, 
until, on mid-summer day, he passes 23° south of it, and 
the day, though of equal length, has less heat than the pre- 
ceding days. 

§ 281. A man at the pole, on the vernal equinox 
grazes the northern edge of the illuminated hemisphere. 
He remains in it, and at noon the next day he has ad- 
vanced south of the spot where he was the preceding day. 

The path of the sun will be parallel to the horizon of an 
observer at the pole. On the day of the vernal equinox, the 
sun will appear half visible above his horizon, and will de- 
scribe a circle around it in twenty-four hours. At the end 
of these twenty-four hours, it will have risen a little above 
the horizon. In the next twenty-four hours, it describes 
another circle, and gains a little in altitude. This con- 
tinues for three months, during which he is in sight all the 
time. He then describes circles, gradually descending, 
till, in three months more, he sinks beneath the horizon, 
for a six months' absence. 

§ 282. A person in latitude 40° north will, on the 
vernal equinox, enter the illuminated hemisphere at right 
angles to its edge ; he will describe an arc which passes 
40° from the centre of illumination, and will leave the 
illuminated hemisphere at right angles after a day of 
twelve hours long. The next day he will enter the illumi- 
minated hemisphere, not at right angles, but obliquely to- 
ward the south, and in a point farther south, will pass a 
part of a degree nearer to the centre, and have a day of 
more than twelve hours. The next day he enters still far- 
ther south and more obliquely, passes nearer the centre of 
illumination, and has a longer day. At the end of three 
months he has passed within 17° of the illumined centre, 
and had his longest day. For the succeeding three months 
the arcs he describes grow less and less oblique to the 
horizon and shorter and shorter, till at the autumnal equi- 
nox he enters the illuminated circle where he did on the 



ELEMENTS OF ASTRONOMY. 199 

vernal equinox, and describes an arc at right angles to the 
circle of illumination. 

For three months after this he enters farther north and 
less and less deeply into the illumined hemisphere, describ- 
ing oblique arcs, until he passes 40° +23° =63° distant 
from the illumined centre, and has his shortest day. 

To such an observer the sun in the vernal equinox would 
rise precisely in the east, pass within 40° of his zenith, 
and set in the west. The next day he would rise five 
sixths of his own diameter north of east, and pass nearer 
the zenith, and set as much further north. At the solstice, 
he would rise only one sixth further north. He would then 
describe his longest arc, and pass within 17° of the zenith. 
After this his arcs would shorten and decline, till in the 
winter solstice the shortest arc would not pass within 63° 
of the zenith. 

The more near a place is to the pole the more oblique 
are the arcs described, and of course the less near they 
approach the zenith of the place. 

A place 10° from the pole enters the illuminated circle 
when the sun is 10° south of the equator, but passes through 
only a small portion of it, and is in it but a few hours. 
Each succeeding day it cuts deeper into the circle, and 
remains in it longer ; and when the sun is 10° north of the 
equator, it remains in the circle all the twenty-four hours, 
describing small circles within it, until the sun has been to 
the tropic and returned to 10° north of the equator. 
Then it describes large, and continually decreasing arcs of 
circles, till at last, when the sun is 10° south latitude, it 
scarcely grazes the illuminated hemisphere. All places 
within 23° 28', and not at the pole, describe in this way 
arcs and circles in the illuminated hemisphere, each with a 
radius equal to its own distance from the pole. 

§ 283. To a person 10° from the pole, the sun is visible 
as soon as he is 10° south of the equator. On the first clay 
he merely appears in the horizon for a short time, coming 
perhaps not half way up the trees. The next day he rises 
much farther toward the east, and describes a higher arch, 
and sets much farther toward the west. Each successive 
day he rises much earlier, and sets much later than the 



200 ELEMENTS OF ASTRONOMY. 

day previous. When he is 10° north of the equator, per- 
petual day begins at a place 10° south of the pole. He de- 
scribes in the heavens circles oblique to the horizon. The 
circles grow higher for three months, and then decline. 
When the sun is within 10° of the equator, the lower edge 
of the circle begins to dip below the horizon. Smaller and 
smaller arcs of oblique circles are described. At length 
the sun merely shows himself in the horizon ; then his place 
is marked only by twilight ; and at last he disappears for 
several weeks. 

That spiral of the sun which appeared like successive 
circles to the observer at the poles, appears like very ob- 
lique and large arcs of circles as he travels south. The 
more obliquely the arcs cut the horizon of any place, the 
more difference of time is there between two successive 
risings of the sun, the more rapidly do the days vary. 
As we approach the tropics the arcs depart more and more 
from the parallel position they had at the poles, and ap- 
proach the rectangular position they had to an observer at 
the equator ; that is, they become less and less oblique to 
the horizon. There is less and less difference in the length 
of successive days, and less and less difference between the 
longest and shortest day of each place. 

§ 284. Having learned that places in different lati- 
tudes are exposed to the sun for different lengths of time, 
we are prepared to find the variations in the length of the 
day in different portions of the globe. Having learned 
that the same place at different parts of the year remains 
more or less time exposed to his rays, we are prepared to 
find the variations in the length of the clay for those 
places where it varies. 

In considering the length of the day, we shall take no 
notice of refraction, twilight, or other causes which influ- 
ence it. 

Since the day of a place at the poles begins when the 
sun enters the northern hemisphere, and ends when he 
leaves it, it is six months in length. 

The shortest clay for a place 5° south of the north pole 
begins when the sun is 5° south of the equator. It is but 
a few hours long. The next day the sun remains above 



ELEMENTS OF ASTRONOMY. 201 

the horizon longer. The days rapidly lengthen. When 
the sun is 5° north of the equator, the longest day to that 
place begins. It lasts some months, until the sun is again 
in 5° north latitude. The days then rapidly decrease in 
length, the last is of the same length as the first, and a 
night equal to the longest day ensues. 

§ 285. Between the pole and the polar circle the 
length of the day changes very rapidly, as may be under- 
stood by considering how each place rises round the pole. 
For every degree which illumination advances northward 
enables very large portions of daily arcs to rise. Of course 
the nearer a place is to the pole, and the longer the longest 
day is, the more rapidly the days of each place must 
change in length. For when the sun crosses the equinox 
the days and nights are equal all over the world. The 
longest day for all places in the northern hemisphere falls 
on the same day of the year ; and as those which are 
nearer the pole have longer days, the increase or diminu- 
tion of each of their successive days must be greater. 
Accordingly within the polar circles which are 23° 28' 
from the pole, we find the greatest range of variation in 
the days. At the polar circle there is one day of twenty- 
four hours, and a night of equal length. Within it the 
longest days are weeks, and as we approach the poles, 
months long. Each place, however, not on the pole itself, 
has some short days. 

Between the polar circles and the tropics, the days are, 
one half of the year, between twelve and twenty -four hours, 
and the other half they are less than twelve hours long. 
As the decrease in length is divided among ninety-one 
days, one quarter of a year, the change from day to day is 
scarcely perceptible. The longest day for all places in the 
northern hemisphere is when the sun is in the northern 
solstice ; the shortest day for all such places is when he is 
in the southern solstice. 

At the equator the days are always twelve hours long, 
for the equator and the circle of illumination being both 
great circles, must always bisect each other. Twice a 
year the sun moves in the equator, as all horizons all over 
the globe bisect his path, and the days and nights are 
equal all over the globe. 



202 ELEMENTS OF ASTRONOMY. 

§ 286. Thus we find that the 4,383 hours of light, 
and the 4,383 of darkness, which make up the year, are 
variously distributed in different parts of the globe. At 
the poles all the hours of light form one period, all those of 
darkness form another. At the equator the 4,383 hours 
of light are divided into 365 equal periods, and those of 
darkness are distributed in the same manner. For all 
places between the polar circles and the equator, the 4,383 
hours of daylight are divided into 365 unequal portions, 
and for places in the same latitude, north or south, the 
division is similar. For places between the polar circles 
and the poles, there are more than one and less than 365 
periods of light. The shortest flay of each latitude is 
equal to the shortest night, and the longest day to the 
longest night. The shortest and the longest day added 
together equal twenty-four or some multiple of twenty-four 
hours, and the number of hours of daylight, in a year, 
equals the number of hours of darkness, for each place on 
the globe. There is, however, a slight exception to this 
rigorous distribution of light and darkness, owing to the 
motion of the earth in her orbit. 

Since the earth's orbitual motion is more rapid when the 
sun is in his southern declination, the days in which he is 
longer below than above the horizon are not for places 
north of the equator quite so numerous as those in which 
he is longer above it, so that the preceding calculation is 
not strictly correct. The motion is greatest when he is 
nearest the earth, and least when he is farthest from her. 
As he is farthest from her nearly at the summer solstice of 
the northern hemisphere, and nearest to her nearly at the 
winter solstice, his motion is slower in our summer than in 
our winter, and the clays vary still more slowly in length 
at the summer than at the winter solstice. For this same 
reason the sun takes longer in describing that half of his 
orbit he is in, in our summer, than that he is in, in our 
winter. In fact, the summer of the northern hemisphere 
is 7 days, 16 hours, and 50 minutes longer than that of 
southern. 

§ 287. The whole number of daylight hours near the 
pole is also increased by the greater amount of refraction 



ELEMENTS OF ASTRONOMY. 203 

in the cold dense atmosphere ; and thus we find that, con- 
trary to what we might expect, the inhabitant of the pole 
enjoys more of the sun's light than the inhabitant of the 
equator. There is some evidence that the polar day is 
prolonged at least a month by refraction. Three Hol- 
landers, who wintered in Nova Zembla, latitude 75°, after 
three months of continual night, saw the sun rise at noon, 
a fortnight sooner than they expected ; this could only be 
owing to refraction. In these cold regions the reflection 
from the ice and snow is very great ; the aurora borealis 
appears also with a splendor quite unknown in 'milder cli- 
mates ; so that without the moon it is frequently so light 
all night that fine print can be distinctly read. The moon 
also rises higher, and is visible longer during the winter 
than the summer months, except at the equator. At 
the poles the winter full moon is visible a fortnight at a 
time, and circles round the horizon like the summer sun. 

The following table shows the length of the longest day 
in each latitude, refraction being allowed for. At the 
polar circles, those which geographers call hour climates 
terminate, and month climates commence. 



Latitude. 


Longest Day 




Hours. 


Minutes. 


7° 18' 


12 


30 


15° 36' 


13 




23° 8' 


13 


30 


29° 49' 


14 




35° 35' 


14 


30 


40° 32' 


15 




44° 42' 


15 


30 


48° 1' 


16 




53° 46' 


17 




57° 44' 


18 




60° 39' 


19 




62° 4' 


20 




6b° 10' 


22 




65° 54' 


24 





204 ELEMENTS OF ASTRONOMY. 



jatit 


btde. 


North 


Latitude. 


South Latitude. 






Continual Daj 


r. Continual Night. 


Continual Day. 


Continual P 






Days. 


Days. 


Days. 


Days. 


66° 


53' 


31 


27 


30 


28 


69° 


30' 


62 


58 


60 


59 


73° 




93 


87 


89 


88 


78° 


6' 


124 


117 


120 


118 


84° 




156 


148 


150 


148 


90° 




186 


179 


178 


177 



§ 288. The duration of twilight has been determined 
very differently by different observers, and in various parts 
of the globe. There are many difficulties in determining its 
length. There is no exact degree of faintness fixed on as 
its close, though it is generally considered to end when the 
smallest stars in the west become visible. Perhaps the 
duration of the morning twilight might be more precisely 
determined than that of the evening twilight, for the eye, 
dazzled by the sun, cannot decide when the last ray of 
light has disappeared. And as the sun between the tropics 
rushes down almost at right angles with the horizon, he 
retains his splendor almost to the last, and the dazzled eye 
is apt to over estimate the succeeding gloom. "Whereas, 
near the poles, the sun descends very obliquely to the 
horizon, dimmed by the lower layers of the atmosphere 
for some time before he disappears. The eye thus grad- 
ually accustomed to the absence of light is prepared to 
over estimate the twilight. Perhaps this consideration ex- 
plains the great diversity of lengths which have been as- 
cribed to twilight, some observers having computed it to 
last until the sun is 6° or 7°, and others till he is upwards 
of 20° below the horizon. 

§ 289. If however we allow twilight to last until the 
sun is 18° below the horizon, its duration in time varies 
exceedingly to observers in different positions. If we recall 
what was said of the apparent motions of the sun to an 
observer on the equator, another at the poles, and a third 
between the two, we shall understand this. 

Let us suppose, round the horizon of each of these ob- 
servers, a band of 18° added, to represent the extent of 



ELEMENTS OF ASTRONOMY. 205 

twilight. Twilight will last till the sun has left this circle. 
The sun is at right angles to the horizon of the man 
at the equator, therefore every degree the sun moves 
will remove him a degree below the horizon. The horizon 
of the oblique sphere he will leave obliquely, therefore 18° 
of his course will not carry him 18° from the horizon, nor 
consequently out of the twilight circle. The more oblique 
the sphere is, or the nearer the observer is to the poles, 
the more degrees the sun must pass through in order to 
quit the twilight circle. 

The shortest possible twilight takes place at the equator 
at the time of the equinoxes. Its duration is one hour, 
twelve minutes. The longest possible twilight takes place 
at the poles. There are there but two twilights in a year ; 
one after the sun has gone below the equator, which con- 
tinues about 39 days, and another, (which, owing to the 
present position of the perihelion, lasts rather longer,) 
before he again crosses the equator. As he is never more 
than 23° below, the period of total darkness is reduced to 
about 100 days. The passage from light to darkness is 
very gradual, and during almost all the twilight there is 
probably light enough for many purposes. 

The longest twilight for places which have the sphere 
oblique takes place in their mid-summer. In latitude 42° 
it is two hours, twenty minutes. In the latitude of Lon- 
don, it lasts all of every night from the 22d of May to 
July 1st. 

§ 290. The variation of seasons is caused by the incli- 
nation of the planes of the ecliptic and the equator, or by 
the non-coincidence of their poles. This cause operates 
both directly and indirectly ; directly by allowing fewer 
rays to fall on a country at one season than another, indi- 
rectly by influencing the length of the periods during 
which they fall. 

The sun is so distant that his rays may be considered 
parallel without any want of exactness. 

Let us consider how the form of the earth influences the 

number of rays received on a given portion of its surface. 

If from the point of the earth which is vertically exposed 

to them, a plane be supposed tangent to that point, and 

18 



206 ELEMENTS OF ASTRONOMY. 

bounded by a circle the size of the equator, this plane will 
intercept all the rays which would otherwise have fallen on 
a hemisphere. They are received on a smaller surface ; 
therefore any measured portion of the plane surface has 
received more than an equal measured portion of the con- 
vex surface. A plane surface exposed at right angles to 
rain is quickly wet, an oblique one receives less rain for 
its size, a vertical one may escape without a wetting. If 
we draw several parallel rays, and cross them at right 
angles by one straight line, and also by a curved line, we 
shall find that it requires the curved line to be longer than 
the straight line to intercept an equal number of rays. 
Now a sphere falls off in every direction from the vertical 
point ; therefore the number of rays received on a given 
extent of surface diminishes very rapidly in proportion to 
the distance from the vertical point. Let us imagine the 
changes in the obliquity of the rays on any particular 
spot, from morning until night, owing to the rotation of 
the earth and the variations in the heat received. Between 
morning, noon and night, a place would receive the sun's 
rays at every degree of obliquity which its latitude allowed. 
In the morning and evening, when the obliquity is very 
great, the heat received is little ; at noon, when the obli- 
quity is lessened, the heat, as we all know, is much 
greater. 

§ 291. The change in the obliquity of the sun's rays, 
owing to the obliquity of the ecliptic, does not in the whole 
year amount to the change in a single day from rotation, 
but it is sufficient to make perceptible variations in the 
heat. 

As the equator is inclined to the ecliptic 23° 28', no 
point more than 23° 28' distant from the equator can ever 
have the sun vertical to it. No country within the tropics 
can ever receive the sun's rays at an obliquity so great as 
46° 5&, and twice during one half the year every such coun- 
try has them vertical. In latitude 42° north we receive 
his rays at mid-summer at an angle of 42° — 23° =19° ; 
in mid-winter, at an angle of 42° -f 23° =65° ; in the equi- 
noxes, at an angle of 42°. At the arctic circle the mid- 
summer rays have an incidence of about 43° ; the mid- 



ELEMENTS OF ASTRONOMY. 207 

winter one of 90°. At the pole the greatest incidence is 
90°— 23°=:67 ; the least is at the equinox=90°. 

We have treated the surface of the earth as if it were 
smooth. The sides of hills receive the sun's rajs at a 
diiferent angle from a plain. The south side of a hill re- 
ceives the sun's rays less obliquely than the plain, and is 
therefore warmer. 

At the equator and at all places between the tropics the 
sun is vertical twice a year. Twice a year in such places 
he casts no shadow. One part of the year the shadows lie 
toward the north, one part of the year toward the south. At 
the tropics, he is vertical once a year. North of the tro- 
pics, he is never vertical, and shadows fall toward the 
north. South of the tropics, they always fall toward the 
south. 

At the equator there are two summers, which are at 
their highest when the sun is in the equinoxes, and two 
winters, one when the sun rises and sets in the north, one 
when he rises and sets in the south. 

At the tropics the summer is hotter than that of the 
equator, because the sun is nearly vertical for several days, 
and the days are more than twelve hours long. The win- 
ter, however, is much colder than any season at the 
equator. 

§ 292. As the longest days at each place must occur 
when the sun is least oblique to it, the two causes of in- 
creased heat act together. Long days give the sun time 
to act. The earth being a rough, dark, opaque body ab- 
sorbs heat. All that portion which is exposed to the sun 
absorbs, all that portion which is turned from it gives out 
heat. Near the poles heat is absorbed uninterruptedly for 
almost six months, and there is an exceedingly hot sum- 
mer ; for another six months heat is given out, and there 
is an extremely cold winter. At the equator, as the days 
and nights are always equal, this second cause of variety 
does not operate, and there is very little change of tem- 
perature. But in other parts of the globe, the great dis- 
parity in the length of days influences greatly the degree 
to which they are heated. As soon as the day, in any 
parallel of latitude, begins to be longer than the night, 



208 ELEMENTS OF ASTRONOMY. 

there is a surplus of heat retained through the night. The 
days continue to increase, the nights to diminish ; the sur- 
plus is more each day, the amount increases rapidly. 

But the clay of greatest heat is not the longest day, nor 
the day of the sun's greatest altitude, any more than the 
hottest hour of the day is twelve o'clock. The earth con- 
tinues to receive more heat than it parts with for some 
time after the summer solstice ; this is added to what was 
before accumulated, and makes the months of July and 
August hotter than June. In the same way it continues to 
part with more heat than it receives long after the winter 
solstice, so that January and February are our coldest 
months. 

§ 293. We can see what the effect of this second 
cause of variety without the former would be, by considering 
the days of a place between the tropics, but not in the 
equator. Such a place has four days in the year in which 
the sun attains at noon an equal altitude. Two of these 
days the sun is north and two south of him. The altitude 
of the sun then would there be no cause of change. But 
on the two days when the sun is nearer the tropic he 
would remain longer above the horizon than the other two. 
Therefore those two days must be hotter than those in 
which the sun passes at an equal distance the other side 
of the zenith. 

The amount of heat parted with at each place, equals 
that received in the course of the year, and thus the mean 
temperature of each place remains unchanged. 

The amounts received in the northern and southern 
hemispheres are nearly equal. We might suppose, that, as 
the earth is nearer the sun in the northern winter than in 
the summer, the northern hemisphere might in the course 
of the year receive more heat than the southern. But 
since the earth moves faster in our winter than in our 
summer, it receives in the course of it no more heat than 
in our prolonged summer, 

§ 294. The ellipticity of the earth's orbit amounts to 
one thirtieth of its mean distance ; therefore the sun's di- 
rect heating power varies one fifteenth; 30 2 — 29 2 =59 ; 
F 5 (y 9 ? =:one fifteenth nearly, This would be sufficient to ex- 



ELEMENTS OP ASTRONOMY. 209 

ao;o;erate the difference of summer and winter in the south- 
ern hemisphere, and to moderate it in the northern. But 
no such effect is produced. For heat diminishes in inten- 
sity according to the inverse proportion of the surface of 
the sphere over which it is spread ; that is, in the inverse 
proportion of the square of the distance. (Plate I. Fig. 4.) 
S A 2 : S M> : : motion at M : motion at A. This is also 
the proportion in which the angular velocity of the earth 
about the sun varies. S A 2 : S M 2 : : heat received at M : 
heat received at A. Therefore equal amounts of heat 
are received from the sun in passing over equal angles 
round it, in whatever part of the ellipse those angles may 
be situated. This is true however the ellipse may be 
divided by the straight line ASP. The two segments, 
AMP and A I P, will however be described in unequal 
times ; but the greater proximity of the sun compensates 
for the more rapid description of the smaller segment, and 
thus an equilibrium of heat in the two hemispheres is main- 
tained. 

§ 295. These two great causes which we have spoken 
of as causing the variety of seasons, are, in some parts of 
the world, counteracted, and in all variously modified. 

In the torrid zone the vertical sun raises such vapors 
and causes such rains, that the season which should be the 
hottest, is in some places the coldest of the year. And 
the intermediate months, which correspond to the spring 
and autumn of temperate climates, are the hottest of the 
year. Between the tropics then we find no regular sum- 
mer and winter, but rainy and dry seasons. 

In polar and circumpolar regions, the days lengthen 
and shorten so very rapidly, that spring and autumn are 
unknown ; vegetation advances with the utmost rapidity, 
and harvests ripen in the short summer, which can never 
be brought to maturity under warmer suns. 

In the temperate zones, the change from summer to 
winter lasts as long as each of these seasons, and we ac- 
cordingly reckon four seasons. These do not however cor- 
respond with the astronomical seasons, for causes before 
given. 

18* 



210 ELEMENTS OF ASTRONOMY. 

The lengths of the astronomical seasons differ considera- 
bly, as the following table shows. 

Days. Hours. Minutes. 

Spring lasts from the vernal equinox to 

the summer solstice, — 92 21 50 

Summer lasts from the summer solstice 

to the autumnal equinox, — 93 13 44 

Autumn lasts from the autumnal equinox 

to the winter solstice, — 89 16 44 

Winter lasts from the winter solstice to 

the vernal equinox, — 89 1 33 

The autumn and winter of the northern hemisphere are 
shorter than the corresponding seasons in the southern, be- 
cause the perihelion is passed through in the northern win- 
ter. If the earth were in its perihelion precisely at the 
time of the winter solstice, the northern autumn and winter 
would be of equal length, and the rest of the year would 
be equally divided between its spring and summer. As 
the perihelion is 10° in advance of the winter solstice, the 
winter season is most shortened by the rapidity of the 
earth in its orbit, and the summer season includes that por- 
tion of its orbit which is performed most slowly. 

§ 296. All animals and plants have periods of repose. 
Some a long arctic sleep, others a slight cessation of their 
energies. All have periods of awakening to which their 
powers and habits are adapted. 

While great causes bring us daily variety, a multiplicity 
of minor and apparently changeful causes bring us some of 
the most stable arrangements in nature. The climate of a 
place is made up of general, innumerable and local causes, 
which blending together, and sometimes counteracting one 
another, give year by year, and even for each month, al- 
most unvarying results. Thus while we repose on the sta- 
bility of this our home, we find daily something new to en- 
joy in its unexpected beauties. 



ELEMENTS OE ASTRONOMY. 211 



CHAPTER XIII. 

POSITION OF PLACES ON THE EARTH AND OF STARS IN 
THE HEAVENS. 

Modes of defining position on the Earth's Surface. Methods of finding 
Latitude. Longitude. Its determination by the Moon's motion. The 
Sextant. Eclipses of Jupiter's Satellites. Determination of Local Time. 
Lunar Distances. The Theodolite. Celestial Globes and Maps. Appa- 
rent Motions of the Planets. The Fixed Stars. The Zodiac. The Con- 
stellations. The Milky Way. Proposed Revision of the Constellations. 

§ 297. Having ascertained the shape and dimensions 
of our globe, we wish to find our position on it. This may 
be done in two ways, by referring our position to the na- 
tural features of land and water, or by giving our latitude 
and longitude. Both modes of description are employed, 
and each has its advantages. Our latitude and longitude 
remain unchanged, and they furnish the shortest and most 
exact mode of describing our position. Latitude gives us 
some notion of the climate of a place. But if we knew 
places on earth only by their latitude and longitude, we 
should find it as difficult to fix them in our mind as to re- 
member the positions of the stars. The natural features 
of the earth are more easily remembered, but their dimen- 
sions, and the latitude and longitude of these, must be 
accurately fixed before we can refer smaller places to 
them. 

No map or chart is of much value as a representation of 
the earth's surface. Particular portions of it may be faith- 
fully represented on a plane surface, but a globe gives the 
only correct idea of it as a whole. There are two modes 
by which a correct representation of the earth's surface 
may be obtained ; by finding the latitude and longitude 
of a great number of points, and filling in the intermediate 
spaces by local surveys ; or by finding the latitude and lon- 
gitude of a few points, two perhaps in each country, and then 
dividing the whole country into a number of triangles. In 



212 ELEMENTS OP ASTRONOMY. 

both of these ways we must refer to the heavens for the 
position of our starting point. 

§ 298. The latitude of a place is easily found. It is 
equal to the altitude of the elevated pole. Equal differ- 
ences of latitude should not however be represented by 
exactly equal intervals of surface, if great exactness is re- 
quired. The ellipticity of the earth causes degrees of 
latitude to be a little longer as we approach the poles. 
Latitude is reckoned from the equator, and is called north 
or south according as the place lies north or south of the 
equator. 

The altitude of the elevated pole above the horizon might 
be directly observed on the limb of the mural circle, if any 
bright star stood directly therein. This not being the case, 
a bright star near the pole, (called the polar star,) is se- 
lected, and observed in its upper and lower culminations — 
that is, when it passes the meridian above and below the 
pole. One half the sum of the star's greatest and least 
altitudes corrected for refraction gives the altitude of the 
pole, and therefore the latitude of the place. 

It may be found by the observed altitude or the observed 
zenith distance of a star or other heavenly body when in 
the meridian. In observations at sea, the sun or moon is 
observed instead of a star, it being difficult, from the mo- 
tion of the vessel, to obtain a correct observation of the 
meridian altitude of so small a body as a star appears. 
On land the inequalities of the surface make it difficult to 
obtain a true horizontal boundary, the zenith distance is 
therefore employed. The declination of the observed star 
being previously known, it must be added to the observed 
zenith distance (corrected), if both bodies are on the same 
side of the equator. But if the place is in north latitude, 
and the star has a southern declination, the declination 
subtracted from the zenith distance gives the latitude. 

The zenith distance of a star may be obtained more ac- 
curately by making several observations on it at different 
altitudes, before and after culminating, when it is near the 
meridian. The latitude may thus be obtained within a 
few seconds. 



ELEMENTS OF ASTRONOMY. 213 

If the latitude of one place is known, that of another 
may be found bj observations on a star which passes near 
the zenith of both places. The calculation is more simple 
when both places are on the same meridian, and when 
both observations are made on the same day. 

§ 299. These operations are so easy in practice, and 
opportunities are so continually offering themselves, that the 
latitude of a place may generally be determined even under 
the most unfavorable circumstances, and its determination, 
by means of celestial phenomena, is the most important ap- 
plication of astronomy to the purposes of civil life. 

But the longitude cannot be so readily found. France, 
Holland and England for a long time offered in vain great 
rewards to any one who should discover a mode of ascer- 
taining longitude at sea. In the latter part of the seven- 
teenth century, Flamstead gave his opinion that if we had 
tables of the places of the fixed stars, and of the moon's 
motions, the longitude might be found. Upon this Mr. 
Flamstead was appointed astronomer royal, and an observ- 
atory was built at Greenwich for him ; and the instructions 
to him and his successors were that they should apply 
themselves with the utmost care and diligence to rectify 
the tables of the motions of the heavens, and the places 
of the fixed stars, in order to find out the so much desired 
longitude at sea, for the perfecting of the art of navigation. 
It was not however till after Mr. Flamstead's death that 
the tables of the moon's motions were corrected, and an 
instrument invented by which altitudes could be taken at 
sea. The principle of this instrument is that property 
of reflected rays by which the angle between the first and 
last directions of a ray which has suffered two reflections 
in one plane, is equal to twice the inclination of the reflect- 
ing surfaces to one another. The instrument is called a 
sextant if one sixth part of a graduated circle is used, a 
quadrant if one fourth part. Sometimes a whole circle is 
used. 

§ 300. Let A B (Fig. 7, Plate I.) be the limb, or 
graduated arc, of a portion of a circle 60° in extent, but 
divided into 120 equal parts. On the radius C B let a 
silvered plane glass D be fixed, at right angles to the plane 



214 . ELEMENTS OP ASTRONOMY. 

of the circle, and on the movable radius C E let another 
such silvered glass C be fixed. The glass D is perma- 
nently fixed parallel to A C, and only one half of it is sil- 
vered, the other half allowing objects to be seen through it. 
The glass C is wholly silvered, and its plane is parallel to 
the length of the movable radius C E, at the extremity E 
of which a vernier is placed to read off the divisions of the 
limb. On the radius A C is set a telescope F, through 
which any object Q may be seen by direct rays which pass 
through the unsilvered portion of the glass D, while 
another object, P, is seen through the same telescope by 
rays, which after reflection at C, have been thrown upon 
the silvered part of D, and are thence directed by a second 
reflection into the telescope. The two images so formed 
will both be seen in the field of view at once, and by mov- 
ing the radius C E, will, (if the reflectors be truly perpen- 
dicular to the plane of the circle,) meet and pass over, 
without obliterating, each other. The motion, however, is 
arrested when they meet, and at this point the angle in- 
cluded between the direction C P of one object, and F Q 
of the other, is twice the angle ACE included between 
the fixed and movable radii C A, C E. The angles : — 

P M Q + M C F = C F D, 

PMQ + MCF = GCD + GCF; 

G C D = R C P, 
RCP=MCF + GCF = GCD; 
PMQ + MCF = MCF + 2GCF, 
PMQ = 2FCG = 2ACE. 

Now the graduations of the limb being purposely made 
only half as distant as would correspond to degrees, the 
arc B E, when read off as if the graduations were whole 
degrees, will, in fact, read double its real amount, and 
therefore the numbers to read off will express not the an- 
gle A C E, but its double, the angle subtended by the 
objects. 



ELEMENTS OE ASTRONOMY. 215 

As the sextant can be held in the hand and requires 
no fixed support, it is of great use in nautical astronomy. 
It not only measures the distance between two stars, or the 
moon and a star, but gives the altitude of the moon or 
stars. For altitudes at sea, as no level, plumb-line, or ar- 
tificial horizon can be used, the sea offing affords the only 
resource. The image of the sun observed, as seen by re- 
flection, is brought to coincide with the boundary of the 
sea as seen by direct rays. Thus the altitude above the 
sea-line is found ; and this corrected for the dip of the 
horizon gives the true altitude of the sun. On land an ar- 
tificial horizon may be used, and the consideration of the 
dip is rendered unnecessary. 

§ 301. Longitude is reckoned from a point arbitrarily 
chosen, and is called east or west according as it is within 
180° east or west of the meridian which passes through 
this point. It is sometimes reckoned westward all round 
the globe, to correspond with the reckoning of right ascen- 
sion. Celestial longitude may be expressed merely in de- 
grees, or in signs, or the sign in which the star lies may 
be mentioned. Thus longitude 45° is either expressed 
thus, or as the 15th degree of Taurus, which is written 
thus, — y 15°, or as Is., 15° ; the Is., or one sign, being 
taken merely as a mode of expressing 30° longitude. 

The longitude of a place is found by means of its local 
time. Every place has its own sunrise, its own noon. All 
places under the same meridian are brought under the 
sun and have their noons at the same time. All places 
not under the same meridian have their noons at different 
times. The astronomer regulates his clock to indicate 
Oh. Om. Os. when the vernal equinox comes on to his 
meridian. He must therefore mention not only at what 
hour an event happened, but at what hour of what local 
time it happened. 

■ Suppose two observers to set and regulate their chro- 
nometers each by his own true sidereal time. If one of 
these chronometers were transported and compared with 
the other, they would differ by the time occupied by the 
equinox, or by any star, in passing from the station of one 
observer to that of another ; in other words, by their dif- 



216 ELEMENTS OF ASTRONOMY. 

ference in longitude expressed in sidereal hours, minutes, 
and seconds. * 

If chronometers were perfect no better mode of ascer- 
taining longitudes than this need be desired. An observer 
provided with such an instrument might, by journeying 
from place to place, ascertain the differences of longitude 
with great precision. 

§ 302. If he travels westward his chronometer will 
appear to gain, though it really goes correctly. Suppose 
he sets out from A, when the equinox is on the meridian, 
or his chronometer at Oh., and in 24 hours sidereal time 
has travelled 15° westward to B. At the moment of his 
arrival there his chronometer will again point to Oh. ; but 
the equinox will be, not on his present meridian, but on 
that of A, and he must wait one hour more for its arrival 
on that of B. When it does arrive there, his watch will 
point not to Oh., but to lh., and will therefore be lh. fast 
on the local time of B. 

If an observer travels westward, and adojpts the local 
time of each place he reaches, he loses an hour for every 
15° he advances. His watch, which shows the number of 
hours which have actually passed, is an hour fast. He 
throws away that hour, and considers that he has travelled 
one hour less than the true number. If he travels entirely 
round the globe, he will have suppressed an hour twenty- 
four times, and thus will reckon one day less than if he 
had remained stationary. Each of his days will have been 
a little longer than to a person at rest, and he will actually 
have seen the sun rise once less often than if he had re- 
mained at home. We have 365 days in a year, because 
the earth carries us round under the sun 365 times a year. 
By travelling westward, one of these turns is gradually, 
for that observer, cancelled, and he has but 364 clays. 
Travelling eastward, a chronometer is found slow ; for 
every 15° of advance the hours are called one more than 
the true number, — one turn is added to the earth's 365 
turns, ono day is gained in a year. This has actually hap- 
pened to navigators. Two settlements on the same meri- 
dian may in Jhis way differ a day in their reckoning of 
time according as they have been colonized by settlers 
arriving from the eastward or from the westward. 



ELEMENTS OF ASTRONOMY. 217 

§ 303. Instead of comparing two local times by a 
chronometer, the instant of the occurrence of & phenomenon 
may be noted in the local time of two places differing in lon- 
gitude, and their difference may be thus ascertained. This 
phenomenon may be a natural or an artificial signal. Natural 
signals, such as eclipses of the moon, may be seen at the same 
instant of time over a hemisphere ; artificial signals are 
visible to a much less distance. The exact local times of twOv 
stations being known, a signal of some definite kind, as a 
flash of powder, is made between the two within sight of 
both. Since light is so very swift, the signal will be seen 
at the same absolute instant at both places, and the differ- 
ence of their local times gives the difference of their lon- 
gitudes. A line of such signals is sometimes used. The 
distance to which they may be seen is very considerable. 
Over the sea the explosion of rockets may be seen fifty or 
sixty miles ; and in mountainous countries at much greater 
distances. Meteors, which are natural signals, may also be 
used to determine longitude. From their great height 
they are seen over a large extent of country, and their 
sudden appearance makes it easy to seize the moment of 
appearance. The magnetic telegraph has also been used 
for finding longitude. Signals are transmitted, apparently 
instantaneously, from one station to another, and the differ- 
ence of the local times of the two stations gives the differ- 
ence in longitude. 

Another natural signal is an eclipse of Jupiter's 
satellites. These eclipses have a great advantage ; 
the time of their occurrence at any fixed station, as 
at Greenwich, can be predicted with such accuracy as to 
stand in the place of a second observation. An observer 
may compare his local time with the predicted Greenwich 
time, and thus learn his longitude. This mode is not how- 
ever susceptible of great exactness, since the moment of 
the commencement of the eclipse cannot be seized ; neither 
can it be employed in navigation, because the rolling of the 
ship prevents nice telescopic observations. 

Lunar eclipses have likewise been used for obtaining 
longitude, but they are liable to the same objection to a 
19 



218 ELEMENTS OF ASTRONOMY. 

still greater extent. An error of a minute in this observa- 
tion would cause an error of a quarter of a degree of longi- 
tude, and in fact a much greater error of observation is 
unavoidable. 

§ 304. The most simple and exact method of finding 
the time at a given place, is to observe the instant when 
the limb of the sun, or a star of known right ascension, 
is on the middle wire of a transit instrument properly ad- 
justed. At that moment the star is on the meridian ; its 
right ascension expressed in hours, minutes and seconds is 
the sidereal time. No method is equal in accuracy to this 
method of transit ; but as it can scarcely be employed ex- 
cept in fixed observations, it is necessary to adopt some 
other more generally applicable. By means of a sextant 
the altitude of a known star may be taken, the time being 
carefully observed, while the star is at a considerable dis- 
tance from the meridian ; when the star has passed the 
meridian, and is at the same altitude on the other side, let 
the time be noted carefully. Since the apparent altitudes 
are equal, refraction is the same, and the true altitudes 
are equal. The instant of the star's being on the meri- 
dian will exactly bisect the interval of the observations. 

§ 305. The method of lunar distances is more use- 
ful than any other in finding the longitude. Its prin- 
ciple is identical with that of lunar eclipses and 
eclipses of Jupiter's satellites. The object is to find 
some celestial phenomenon which may be observed under 
diffei^ent meridians, and by which the two observers 
may compare the times they reckon at the same absolute 
instant. The hour at which the phenomenon will happen 
may be calculated for one meridian, and observed for the 
other, and the comparison of the observed with the calcu- 
lated time is substituted for that of two observed times. 
The face of the heavens has been compared to the face of 
a dial, on which well known bright stars are the marks 
which give the time, the moon is the hand, and Greenwich 
time is 12 o'clock, or the starting point. The marks on 
the dial are not at equal intervals, but the intervals are 
known ; the moon varies in rapidity of motion, but her 



ELEMENTS OF ASTRONOMY. 219 

variations are known ; the observer does not stand in the 
same plane with the centre of the moon and the centre of 
the earth, but he can allow for the change this causes in 
the moon's apparent place. The moon is selected for the 
index rather than the sun or any planet, because her 
greater rapidity of motion leaves less uncertainty about 
the precise moment corresponding to any given angular 
distance from the stars. The sun's apparent motion is 
only one thirteenth of that of the moon ; and if his place 
were determined with an uncertainty of only a quarter of a 
minute in space, this would leave an uncertainty of nearly 
six minutes of time in the longitude. 

§ 306. It is evident that there is only one particular 
instant at which the moon can be at a certain distance 
from any fixed star. If this distance is ascertained at any 
moment, and the Greenwich time at the moment she had 
the same difference is counted from the tables, the longi- 
tude is found. The true distance of the moon's centre 
from the star, when corrected for refraction and paral- 
lax, is the same for every meridian. The British Lords 
Commissioners of the Admiralty publish annually a Nauti- 
cal Ephemeris, containing the distance between the moon 

and certain bright fixed stars near her path, for every three 
hours. 

We have thus, in lunar distances and in chronometers, 
two independent methods of finding the longitude, each of 
which may act as a check on accidental errors in the other. 
Several chronometers may likewise be used to correct one 
another. The rate at which a chronometer gains or loses 
should always be verified before leaving some known meri- 
dian. And if a vessel remains long enough in port, its 
rate should again be ascertained, since change of tempera- 
ture or other causes may have altered it. If the rate is 
accurately known, and the error allowed for, chronome- 
ters will give the longitude throughout a long voyage. 

§ 307. In determining the relative position and dis- 
tances of places on land, use is made of an instrument 
called a Theodolite, which resembles the altitude and 
azimuth instrument ; its use has been thus described. 
It is evident that, as every object to which the tel- 



220 ELEMENTS OF ASTRONOMY. 

escope of a theodolite is pointed has some certain ele- 
vation, not only above the soil, but above the level of 
the sea, and as, moreover, these elevations differ in 
every instance, a reduction to the horizon of all the 
measured angles would appear to be required. But, 
in fact, by the construction of the theodolite, this re- 
duction is made in the very act of reading off the hori- 
zontal angles. Let E (Fig. 5, Plate I.) be the centre 
of the earth; A, B, C, the places on its spherical sur- 
face, to which three stations, A, P, Q, in a country, are 
referred by radii E A, E B P, E C Q. If a theodolite be 
stationed at A, the axis of its horizontal circle will point to 
E, when truly adjusted, and its plane will be a tangent to 
the sphere at A, intersecting the radii EBP, E C Q, at 
M and N, above the spherical surface. The telescope of 
the theodolite, it is true, is pointed in succession to P and 
Q ; but the readings off of its azimuth circle give, not the 
angle P A Q, between the directions of the telescope, or 
between the objects P, Q, as seen from A ; but the azi- 
muthal angle M A 1ST, which is the measure of the angle A 
of the spherical triangle B A C. The sum of the three ob- 
served angles of any of the great triangles in geodetic 
operations is always found to be rather more than 180° ; 
and this excess, which is called the spherical excess, is so 
far from being a proof of incorrectness in the work, that it 
is essential to its accuracy, and offers at the same time 
another palpable proof of the earth's sphericity. 

§ 308. The true way, then, of conceiving the subject 
of a trigonometrical survey, when the spherical form of the 
earth is taken into consideration, is to regard the net-work 
of triangles with which the country is covered, as the bases 
of an assemblage of pyramids converging to the centre of 
the earth. The theodolite gives us the true measures of 
the angles included by the planes of these pyramids ; and 
the surface of an imaginary sphere on the level of the sea 
intersects them in an assemblage of spherical triangles, 
above whose angles, in the radii prolonged, the real sta- 
tions of observation are raised, by the superficial inequali- 
ties of mountain and valley. These triangles may after- 
ward be reduced to the level of the sea, by applying the 



ELEMENTS OF ASTRONOMY. 221 

rule for the spherical excess, and the ellipticity of the 
earth may also be taken into account in very nice surveys. 

The irregularities of the earth's surface are learned by 
sounding the sea and by applying the barometer to the air, 
or by direct measurement of heights. The pressure on the 
barometer at any height informs us in what stratum of the 
air that height is, and consequently how much it is above 
the sea level. 

§ 309. Celestial globes are more easily constructed 
than representations of the earth. A concave surface 
would of course be the most perfect representation of the 
heavens, but as we cannot have one so large that the spec- 
tator can stand within it, convex globes are generally used. 
Triangles may be transferred from the heavens to this 
surface, bright stars taking the place of stations in terres- 
trial triangles. A better way to construct a globe is, as 
the earth rotates to observe the place in the heavens of 
each celestial object which passes our meridian, and to 
refer it to its place on an imaginary sphere conceived to 
revolve with the stars. By observing both in a north 
and south latitude the whole sphere may be mapped 
out, and their true places assigned to the fixed stars. 
As on the earth's surface we may refer points to the 
natural portions of the earth, or to latitude and longi- 
tude, so in the heavens we may describe a star by its 
situation relative to others, by its belonging to some 
constellation, or by its right ascension and declination. 
The features of the earth probably change in long ages, 
continents rise and fall, mountain chains are thrown up, 
gulfs open. We have learned that apparently slight 
changes constantly occurring may, in equally long or per- 
haps longer intervals, alter the appearance of the heavens. 

Terrestrial latitude and longitude we have every reason 
to suppose immutable. But this is not the case with de- 
clination and right ascension. As on the earth the meri- 
dian of Greenwich or of some other place is selected as 
the point from which longitude is counted ; so in the 
heavens, the vernal equinox, one of the points in which 
the equator and ecliptic intersect, is chosen for the zero 
19* 



222 ELEMENTS OF ASTRONOMY. 

point of right ascensions. If the axis of the earth is, as 
we have called it, absolutely immovable, this point of in- 
tersection will always remain the same, and right ascen- 
sions and declinations, celestial latitudes and longitudes, 
will remain unchanged. If by any cause motion of the 
earth's axis is induced, the equinoctial must share this 
motion, and must intersect the ecliptic in a different place. 
Or if the position of the ecliptic among the stars should, 
by any cause, be changed, the points of intersection would 
in like manner be moved. This leads us to the inquiry, 
which will be answered in the next chapter, whether there 
are in the heavens any perturbations. 

§ 310. Maps* of the heavens constructed by an in- 
habitant of the earth, will be in their natural features cor- 
rect for an inhabitant of any of the planets, but not in their 
arbitrary references. The constellations and the milky-way 
are the same to all dwellers in our system, but each planet 
would have its own ecliptic and equator, its hour-circles, 
and lines of longitude to which to refer stars. 

We have assigned no place on the celestial globe to comets, 
to the planets, or to the sun and moon, which often, from 
their apparent wandering, are included among the planets. 
Since the planets change place continually, their position is 
better described by giving their distance from the sun. 
As the sun is the centre of our system he could not very 
well appear on a map or globe, and as we transfer our 
motion to him he also occupies different positions in the 
heavens. Perhaps it will be well to complete our map of 
the heavens by mentioning to what portions of it the planets 
are confined, and how their motions in that portion appear. 

§ 311. Owing to the earth's motions, the sun, as seen 
from the earth, covers in the course of the year the celes- 
tial zone, extending from 23° 28' north declination, to 
23° 28' south declination. It covers every portion of this 
zone in the course of the year, its path one day partially 
lapping over its path the preceding day. It never departs 
from this zone north or south. 

The moon really moves round the earth in a kind 
of spiral, so that her disc at different times passes over 
every point in a zone of the heavens, extending rather 



ELEMENTS OP ASTRONOMY. 223 

more than 5° 9' on each side of the ecliptic. At one time 
or other she occults every planet and star within this space. 
The occultation of a star by the moon is not more frequent 
than by the sun, but the dimness of the moon allows it to 
be seen. The moon seems to pass over the star, which in- 
stantaneously vanishes at one side of her disc and after a 
short time as suddenly reappears on the other. 

The sun and the moon are so important to the earth 
that the daily apparent course which the earth's rotation 
impresses on them is noticed. This is from east to west, 
while the moon's actual monthly and the sun's apparent 
yearly course are from west to east. The earth likewise 
impresses daily motions from east to west on the planets, 
but these are left out of consideration in tracing the courses 
of the planets. 

No eye but one placed entirely outside of the system, 
or at the sun, could see the motions of the planets truly. 
Since their paths are nearly in the plane of the ecliptic, 
we see their motions not in plane but in section ; their real 
angular movements and linear distances being all fore- 
shortened and confounded undistinguishably, while only 
their deviations from the ecliptic appear of their natural 
magnitude, undiminished by the effect of perspective. 

To an observer whose point of view is itself in motion, 
the paths of the planets are transformed into zigzag lines ; 
they appear now to advance rapidly, now to stand still, and 
then to recede. The planets nearer to the sun than the 
earth are called inferior, those more distant than the earth 
are called superior planets. There are some differences in 
the motions of the superior and the inferior planets as 
viewed from the earth. 

§ 312. The inferior planets appear to vibrate each 
side of the sun, never removing far from him, and advanc- 
ing with him in the ecliptic. An inferior planet has two 
conjunctions, a superior one when it is beyond the sun 
with regard to the earth, and an inferior one when it is 
between the sun and the earth. Superior planets have 
one opposition and one conjunction. "When an inferior 
planet passes beyond the sun, with regard to the earth, 
its motion, compared to the stars, appears direct. When 



224 ELEMENTS OF ASTRONOMY. 

it passes between the earth and the sun, its motion appears 
retrograde. In transition from one of these states to the 
other, its motion is imperceptible, and it appears stationary. 
These appearances would take place if the earth were sta- 
tionary. The earth's motion only modifies them by chang- 
ing the points at which they become stationary, and by 
making them appear stationary longer. Conjunction also 
takes place less frequently in consequence of the earth's 
moving in the same direction with the planets. While the 
planet has performed half of its orbit, the earth has ad- 
vanced, and the planet must also advance to be in conjunc- 
tion with the sun. Thus conjunction of the inferior planets 
takes place rather less than twice in one of their years. 
The planet appears to vibrate a few degrees on each side 
of the sun, but never appears in any remote part of the 
heavens. 

§313. The superior planets appear more capricious 
in their movements. Let us suppose ourselves on a great 
lake in a boat, while another boat passes in the same direc- 
tion between us and the shore. If both move with equal 
quickness, the boat nearest the shore will appear motion- 
less. If the shore is at a great distance, the external boat 
will answer for a long time to the same objects. If one of 
the superior planets represents the boat nearest the shore, 
and the sphere of the fixed stars an infinitely distant bank, 
while the earth and this planet move with equal quickness 
and in parallel directions, the latter will appear to us to 
answer to the same point of the heavens, to be stationary. 
If the boat nearest the shore goes less swiftly, the spectator 
in the other will see it retrograde, and hide successively 
along the shore objects in the direction opposite to that in 
which he goes. When, on the contrary, the former goes 
more quickly, it will appear to advance in the true route 
and directly. If the one farther from shore moves in an 
opposite direction, the nearer motion will still be direct and 
very rapid. All this represents what happens to the supe- 
rior planets with regard to the earth. When the latter 
surpasses them in quickness in the same direction, they 
are retrograde ; they become direct when the earth is left 
behind or is in that part of her orbit which has a contrary 



ELEMENTS OF ASTRONOMY. 225 

direction. Thus the stations and retrogradations of the 
superior planets are toward oppositions ; they are direct in 
the rest of their course. 

§ 314. When a superior planet approaches its opposi- 
tion there must be a little arc where two lines joining the 
ends of the planet's arc with the ends of the earth's are 
parallel. During this time the planet is stationary, for be- 
ing referred to the immensely distant fixed stars, the dis- 
tance between the two ends of the arc is not perceived, 
and the two positions are confounded. Afterward the 
earth surpasses the planet in quickness, its motion becomes 
retrograde, and opposition is made toward the middle of 
the arc of retrogradation. At last the retrograclation is 
stopped by the planet's becoming stationary. It then be- 
comes direct, and remains so as long as the earth passes 
through that part of her orbit where she moves in an oppo- 
site direction from the planet. The motion of the planet 
accelerates continually from its last stopping-place until it 
is hidden in the rays of 'the sun, and from the moment 
when it disengages itself, it slackens its swiftness till the 
next station. 

The motion of the earth alters tile motions of the supe- 
rior planets far more than those of the inferior. If it were 
at rest the planet would mark out on the sphere a circle, 
and have one opposition and one conjunction in each of its 
own years. As it now is, Jupiter, in each of his revolu- 
tions, is twelve times overtaken and passed by the earth, 
Saturn thirty times, and they have respectively twelve and 
thirty oppositions and conjunctions. Their paths among 
the stars are zigzags, retrograding as many times as the 
planet has oppositions in a year. Thus their apparent 
often far outstrips their real motion. Herschel does not 
move through 5° of his orbit in a year, yet to us he often 
appears to move much more rapidly. On the w T hole, how- 
ever, the amount of direct motion more than compensates 
for the retrograde ; and by its excess the gradual advance 
of the planet from west to east is maintained. Saturn ap- 
pears longer in turning than Jupiter ; Jupiter than Mars ; 
Mars than Venus. For a week together the most power- 
ful telescope scarcely shows any change in the situation 



226 ELEMENTS OF ASTRONOMY. 

of Saturn ; during a whole month he moves less than a 
degree. 

It rarely happens that more than one or two planets are 
in the same part of the heavens at the same time. More 
than 2,500 years before our era, the five great planets 
were in conjunction. 

§ 315. The appearances of the planets convinced 
Copernicus that the earth was not, as then believed, the 
centre of the system. He observed that Mars and Jupi- 
ter and Saturn appeared much larger in their oppositions 
than in their conjunctions. Thus they either had not the 
earth for a centre, or had prodigious excentricities. If 
the sun were the centre, the change of size would be sim- 
ply accounted for. He would not publish his discovery till 
he had satisfied himself that it accounted for all the details 
of their motions. After thirty-six years, with great reluc- 
tance, he gave it to the world, not as a physical truth, but 
as a convenient hypothesis. 

Not only the variations in size, but all apparent irregu- 
larity of the planets' motions vanishes as soon as they 
are referred to the sun as a centre. Their periodic times, 
their distances &nd velocities, in short all the elements -of 
their orbits may be found. All these must be found to 
give the position of a planet in space ; finding its latitude 
and longitude only give its place (among the constellations) 
on the celestial vault. 

§ 316. The latitudes and longitudes mentioned thus 
far have been reckoned from the earth as a centre, and for 
the fixed stars they are the same as if reckoned from the 
sun as a centre. But the places of the sun, moon and 
planets referred to a sphere having the earth for its centre 
differ from their places on a sphere concentric with the 
sun. Their longitudes and latitudes found on the former 
sphere are called geocentric, those on the latter sphere are 
called heliocentric. If the sun is the centre, the longi- 
tudes are calculated from the vernal equinox. The sun's 
and the earth's latitude are always nothing. The helio- 
centric longitude of the earth equals the sun's geocentric 
longitude-}-180 o , or vice versa. The heliocentric equi- 
noxes and solstices are therefore the same with the geocen- 
tric. The geocentric place of the moon or of one of the 



ELEMENTS OF ASTRONOMY. 227 

planets being known, and the earth's distance from the sun 
and from the moon or planet being known, the moon's or 
planet's heliocentric place may be found. Or if you wish 
to calculate the geocentric place of a planet, let its helio- 
centric place and that of the earth be known ; the angle at 
the sun is the difference between these two, and complet- 
ing the triangle we have the place of the planet with re- 
gard to the earth. Sometimes the heliocentric longitude 
precedes the geocentric, sometimes it follows it, and some- 
times coincides with it.. 

If the earth moved in a circle with a uniform velocity 
about the sun placed in the centre, its position at any time 
with regard to the line of equinoxes could easily be calcu- 
lated ; for as one year is to the time elapsed, so would 
360° be to the arc passed over. The longitude so calcu- 
lated is called the mean longitude.- But since the earth's 
orbit is neither circular nor uniformly described, this rule 
will not give us its true place in the orbit at any particular 
moment. Still the true place differs very little from the 
place so determined, called the mean place, and may al- 
ways be found from it by applying a correction, or equation 
as it is termed, whose amount is not very great, and de- 
pends on the equable description of areas about the sun. 
The proportion is ; as one year : the time elapsed : : the 
whole area of the ellipse : the area of the section swept 
over by the radius vector in that time. 

§317. The quantity by which the true longitude of 
the earth differs from the mean longitude is called the 
equation of the ceDtre, and is additive from perihelion to 
to aphelion, beginning at 0°, increasing to a maximum, and 
again diminishing to zero at the aphelion. After this it 
becomes subtractive, attains a maximum, and again dimin- 
ishes to 0° at the perihelion. Its maximum, both additive 
and subtractive, is 1° 55' 33". 

• The maximum value of the equation of the centre de- 
pends only on the ellipticity of the orbit. If then the former 
inequality can be ascertained from observation, the latter 
may be found from it. The sun's exact longitude may be 
ascertained for every day, and compared with the mean 
longitude, and the greatest amount of its defect or excess 



228 ELEMENTS OE ASTRONOMY. 

ascertained. This is a more accurate mode of learning the 
excentricity of the orbit than that of concluding its dis- 
tance from its apparent diameter. Since the true and 
mean longitudes agree twice a year, but do not agree from 
day to day, the true must, during a part of each half year, 
increase more rapidly, and during a part less rapidly than 
the mean. The earth, starting from perihelion, describes 
each day arcs which exceed 3^-5 part of its orbit ; as it ap- 
proaches the aphelion, the arcs described fall short of -3-^ 
of its orbit. But the true longitude each day measures the 
earth's whole advance from the perihelion. If it gains a 
great deal the first day, and less the next, and less the one 
after, and so on, the whole amount added to the mean lon- 
gitude will for some time increase, but a time must come 
when it will diminish ; since the orbit is nearly circular, 
this point is near quadrature. We shall find hereafter 
that the longitude of the earth's perihelion has a very slow 
advance on the ecliptic. This of course causes the equa- 
tion of the centre to be additive in different portions of the 
ecliptic in different centuries. 

§ 318. From whatever spot on earth we look into the 
heavens, we can never, even with telescopes of the highest 
power, discover any limit ; nothing intervenes to check our 
sight. The heavens assume the form of a hollow sphere, 
of which our eye is the centre, because the eye seeing 
equally far in all directions, refers all it beholds to a con- 
cave surface every where equally distant from itself. 
A hollow sphere is the only surface which answers this 
condition, and the eye maps the stars clown on this sphere 
with reference merely to their direction from us ; the 
faintest may be millions of miles beyond -the brightest ; 
their actual distance from us does not affect their place on 
the sphere. In the same way, when we are painting a 
group of persons, each one takes his place on the back- 
ground formed by the wall without regard to his distance 
from us ; and just so we refer all the buildings, trees and 
figures in a landscape to the mountain which closes in our 
horizon. 

But when we look into space, there is no mountain, no 
wall, to shut in our horizon ; our view is only limited by 



ELEMENTS OF ASTRONOMY. 229 

the extent of our power of vision. This power differs ex- 
tremely in different persons, and no one is conscious where 
it ends, though all are conscious of the apparent shape 
which arises from its ending. The radius of our sphere of 
vision includes the milky light of stars in other clusters. 
All things within this distance, whether thousands, millions, 
or millions of millions of miles distant, are referred to one 
same imaginary surface. Only one half of this sphere is 
visible to one observer — the earth on which he stands ex- 
cluding from view the hemisphere beneath his feet. By 
means of observations in different latitudes, and at all sea- 
sons of the year, the whole surface of this concave sphere 
is however familiar to us, and is recognized by means of 
its natural features, the stars. 

§ 319. But though our only landmarks, the stars are 
at first view like the sands of the desert or a snowy plain. 
We are however not wholly without signs. We notice 
striking differences in the size, color and arrangement of 
the stars among themselves ; we find their arrangement 
never varies ; that stars which have to our fancy taken the 
form of a bear, retain it whether the bear have his feet or 
his head toward our horizon. We begin to map out the 
heavens, and since we find nothing like those outlines of 
land and water which define the surface of our globe, we 
invent, or rather some Chaldean shepherds invented for us, 
some thousands of years ago, odd, uncouth figures which 
bear a slight or fancied resemblance to the forms of the 
groups of stars. Wherever a few bright stars lay suffi- 
ciently near together to assume the form of a Lyre, a Swan, 
a Virgin, or any other figure, they were called a constella- 
tion, and took their name from their fancied form. Thus 
in time the whole celestial vault has been covered with 
imaginary figures which serve well enough to indicate to 
the common inquirer in what part of the heavens a star is 
placed, and which have usually some mythological or his- 
torical interest which aids the memory. Probably the 
most striking stars and groups were the first to receive a 
name. The natural event most important in its effects, 
and most obvious to an uncultivated people, is the return 
of the seasons. This must have been early perceived to 
20 



230 ELEMENTS OF ASTRONOMY. 

be accompanied by an apparent motion of the sun among 
the stars. If the light of the stars were much stronger, or 
that of the sun much weaker, we might see him pass by 
the stars in each part of the ecliptic, as we do the moon. 
But his path was easily ascertained by observing what stars 
rose or set with him each night, or what were opposite to 
him at midnight. 

§ 320. It must have been early observed that the 
planets and the moon never wandered far from the ap- 
parent path of the sun. All the motions of the planets 
then discovered, are performed within a zone extending 
about 8° each side of the earth's path. The stars in this 
zone were very early formed into constellations, and con- 
sidered as resembling figures of animals. The name 
Zodiac, from a Greek word signifying animal, was conse- 
quently applied to this zone. The constellations which lie 
in it are by us called the zodiacal constellations. It has 
been supposed that the country and period in which they 
were named might be ascertained by calculating in what 
country the agricultural operations of which the signs are 
symbols would coincide with the presence of the sun in 
that constellation. Some antiquarians have inferred that 
Egypt is the country ; but as the Egyptians borrowed 
their mythology and perhaps their civilization from some 
Oriental people, it seems more probable they received their 
astronomical calendar from the same source. And as the 
Hindoo and other Oriental nations show some traces of a 
similar division, it is probable that they and the Egyptians 
received them from the same more ancient source. Per- 
haps the Egyptians received the division as a loose one, 
and first made it definite ; for several coincidences make 
it probable that it received its present arrangement in 
Egypt. 

Owing to causes hereafter to be explained, the sun does 
not appear among the same stars at the same season in 
which he did centuries ago. If we consider the zodiacal 
constellations as symbols connected with husbandry, the 
agreement could only have subsisted when the sun was in 
the constellation Aries on the 21st of March. 

§ 321. We may then suppose the constellations Aries, 



ELEMENTS OF ASTRONOMY. 231 

Taurus and Gemini, to have been named from the young 
of animals being added to the flock in spring. After this 
the sun seems to retreat toward the south, and the next 
constellation is called Cancer, from the crab, which moves 
backward. Leo indicates the violent heats of summer, and 
Virgo represents a gleaner, and the time of her appearance 
coincides with harvest time in Egypt. The perfect equal- 
ity of the days and nights in the next month is symbolized 
by Libra, the balance. The diseases produced by the de- 
parture of the sun gave to the next sign the name of Scor- 
pion, because it is mischievous, and was thought to have a 
sting in its tail. The next month was the season for hunt- 
ing, and had for its emblem Sagittarius, an Archer. In 
the next month the sun appeared to ascend from the south 
toward the equator, and it had for its emblem Capricornus, 
because the goat is accustomed to ascend the highest points 
of ground. The next sign is Aquarius or the Watercarrier, 
named from the rains that generally fall at this season, or 
from the inundation of the Nile. And the last sign is 
Pisces, the Fishes, so called perhaps because they were 
thought to be most fit for use at that time. 

If the names were given to the constellations in which the 
sun then was, the antiquity of 15,000 years is required for 
the zodiac. Perhaps they were given not to those in which 
the sun then was, but to those which were opposite to him, 
and which consequently were rising at sunset at any given 
spot. This theory brings down the invention of the con- 
stellations to about 2,500 years before Christ ; it has been 
adopted by La Place and several distinguished philoso- 
phers. 

The Greeks probably received their astronomical knowl- 
edge from the Egyptians. It is evident they did not them- 
selves name the constellations, because they could not, for 
some time at least, explain them according to their own 
mythology. Probably out of the medley of men, animals, 
and other objects, with which earlier astronomers had filled 
the heavens, they selected and retained the figures which 
suited the deeds of their own heroes and deities. Thus 
Aries is supposed to represent the ram, whose golden fleece 
was the object of the Argonautic expedition ; Taurus, the 



232 ELEMENTS OF ASTRONOMY. 

bull which was tamed by Jason ; Gemini, Castor and Pol- 
lux. The ship among the southern constellations is sup- 
posed to be the Argo ; and the Ursa Major, which to the 
Greeks would never set, is the nymph Calisto, whom 
Juno forbad Oceanus to receive into his bosom. When the 
Scorpion appears in the east, Orion must sink beneath the 
western horizon, because Artemis, to punish the audacity 
of the mighty hunter, sent a scorpion which bit him in the 
heel. 

§ 322. When it was found convenient to divide the 
ecliptic into twelve equal parts of 30° each, it was found 
that each of these parts would be in the neighborhood of 
one of the zodiacal constellations. For these are twelve 
in number, and situated at nearly equal distances one from 
another. The ecliptic was thus divided ; and each portion 
was named for the constellation which was near it ; and the 
divisions themselves, and also the constellations which gave 
name to them, were called the signs of the zodiac, and 
characters were invented to express them. The names of 
the constellations, or signs, and the characters used to ex- 
press them, are as follows : — Aries, or the Ram, °f > 
Taurus, or the Bull, y ; Gemini, or the Twins, n ; 
Cancer, or the Crab, Z5 ; Leo, or the Lion, ft, ; Virgo, or 
the Virgin, Tt£ ; Libra, or the Balance, £1 ; Scorpio, or 
the Scorpion, v\ ; Sagittarius, or the Archer, f ; Capri- 
cornus, or the Wild Goat, V? ; Aquarius, or the Water- 
carrier, zz ; and Pisces, or the Fishes, X . 

When these constellations were fixed on to determine 
the names of the subdivisions of the ecliptic, the vernal 
equinox was a point very near the Bam, the summer sol- 
stice was near the Crab, the autumnal equinox was near 
the Balance, and the winter solstice near the Wild Goat. 
The vernal equinox therefore received the name of the 
first point in Aries, which it still retains ; the autumnal 
equinox was the first point in Libra ; and the tropics were 
called the tropics of Cancer and Capricorn. These names 
still continue in use, though the circumstances from which 
they took their origin have ceased to exist. The vernal 
equinox now takes place in the constellation Pisces, the 
autumnal equinox in Virgo, the summer solstice in Gemini, 



ELEMENTS OF ASTRONOMY. 233 

and the winter solstice in Sagittarius ; and the other con- 
stellations from which the signs are named, have also 
changed their situation on the circle of the ecliptic. Not 
only, however, do the points of the equinoxes and the sol- 
stices retain their names, but the whole ecliptic is still 
divided into twelve portions, which are called signs, and 
retain the names of the constellations for which they were 
originally called. These signs, or portions of the ecliptic, 
continue to be measured at intervals of 30° each, from the 
actual position of the vernal equinox ; the equinox having 
retreated, the signs or constellations of the zodiac no 
longer correspond with them. The constellation of the 
Earn is now near the sign $ of the ecliptic ; that of the 
Lion near w% ; that of the Waterman near X . Hence we 
must carefully distinguish between the signs of the zodiac, 
which are fixed with respect to the equinoxes, and the con- 
stellations, which are movable with respect to these points. 

The ancient Greeks reckoned only forty-six or forty- 
seven constellations. Hipparchus added Equuleus. The 
Hair of Berenice and Antinous afterward made the number 
fifty. 

§ 323. In the fifteenth century, when navigation was 
extended beyond the equator, and sailors noted those stars 
in the southern hemisphere which were not visible to the 
ancients, they found it convenient to group them into con- 
stellations. They did not however adapt them to the 
Greek mythology, but selected principally such objects as 
presented themselves in the newly discovered countries. 
Whence we have for the southern constellations, the Phoe- 
nix, the Toucan, the Little Water Snake, the Sword-fish, 
the Flying-fish, the Fly, the Chameleon, the Bird of Para- 
dise, the Peacock, the Indian, and the Crane. 

The ancients took only those parts of the heavens as the 
ground-work of the constellations where the bright stars 
existed. Consequently in many places there were no con- 
stellations, and the stars which were scattered over such 
places were called in formes. Some of these empty spaces 
were very great, and contained here and there stars which 
were as much entitled to be formed into constellations as 
those of several existing ones. Therefore modern astrono- 
20* 



234 ELEMENTS OF ASTRONOMY. 

mers named the new constellations, called the Camelo- 
pard, the Unicorn, the Fly, and the rivers Jordan, Eu- 
phrates, and Tigris. In the latter part of the seventeenth 
century, the rivers were rejected, and instead of them, and 
in some other vacant spots, were introduced the Hounds, 
Mount Menalus, Cerberus, the Fox and Goose, the Lizard, 
the Shield of Sobieski, the Lynx, the Little Lion, the Little 
Triangle, and the Sextant, and also the Bow and Arrow of 
Antinous. 

Many other names of constellations were added as 
compliments to monarchs or patrons, or as commemora- 
tive of interesting events or distinguished men. These 
often replaced former constellations, and speedily disap- 
peared from the maps, and many are not known at the 
present day. 

§ 324. A natural feature in the heavens, more marked 
than any of the constellations, is the milky-way. This has not 
improbably presented the same appearance and kept the same 
position ever since the creation of our cluster. It traverses 
the constellations Cassiopeia, Perseus, Auriga, Orion, 
Gemini, Canis Major, and Argo, where it appears most 
brilliant. It then passes through the feet of the Centaur, 
the Cross, the Southern Triangle, and returns towards the 
north by the Altar, and the tail of the Scorpion, where it 
divides into two branches. One branch passes through 
the tail of Scorpio, the bow of Sagittarius, Aquila, Antinous, 
Sagitta, and the Swan. The other branch passes through 
the upper part of the tail of Scorpio, the side of Serpenta- 
rius, Taurus, Poniatowski, the Goose, and the neck of the 
Swan, where it again unites with the other branch, and 
passes on to the head of Cepheus. Here the branches 
unite, after remaining separate for the space of more than 
100°. There is another small separation of the milky-way 
between Cassiopeia and Cygnus. In some parts this zone 
is ten or fifteen degrees broad, as in the southern parts of 
Scorpio, Ara, and the Cross ; in others, as between Per- 
seus and Auriga, it is not more than five degrees in width. 
Some parts of it are visible at all seasons of the year. In 
northern latitudes it is most conspicuous from July to No- 
vember. It is most brilliant in the southern hemisphere. 



ELEMENTS OF ASTRONOMY. 235 

Instead of a confused milky light, it is there more studded 
with brilliant stars. 

§ 325. One would suppose that nearly eighty con- 
stellations were quite enough for all useful purposes. But 
in the eighteenth century twenty-six more were added to 
the number. This extravagant number of new constella- 
tions, some of which were formed of scarcely visible stars, 
by no means made the study of astronomy more easy, but 
on the contrary confused it, and rendered it difficult. More- 
over the new constellations are unsuited to the others, and 
chosen without taste. Astronomical instruments have some 
claim to a place in the heavens ; but figures like the 
Chemical Furnace, the Easel, the Air-pump, the Printing 
Press, and the Electrical machine have no natural relation 
to the sky. 

It is desirable that the heavens should be freed from so 
tasteless and useless an accumulation. In doing this, uni- 
formity must be secured for the maps. The same constel- 
lations must be retained in all, and the same stars should 
be placed in the same parts of the figures. The forms 
chosen should be beautiful and pleasing, their outlines 
should be definite, and when once adopted should remain 
unchanged. It would be well to avoid similarity of names 
in the constellations. We have now an Ursa Major and 
Minor, three Triangles, Pisces and Piscis, Telescopium re- 
peated three times, &c. 

§ 326. According to the present system, some con- 
stellations are so extensive that they exhaust three or more 
alphabets, and therefore it is necessary, beside the letter 
of the star, to give its right ascension and declination. 

The largest stars of each constellation are named by the 
letters of the Greek alphabet, beginning with the brightest 
and proceeding in order. The stars next in brightness are 
numbered according to the Roman alphabet, and sometimes 
a third alphabet of Italian letters, or one numbered is re- 
quired, as a 2 ; or numbers alone are used. 

The letters do not indicate the magnitude of the stars 
they represent, but merely the relative magnitude of those 
in the same constellation. Thus a Virginis is a star of the 



236 ELEMENTS OF ASTRONOMY. 

first magnitude ; a Librae, a star of the second magnitude ; 
and 73 Aquarii, a star of the third magnitude. 

Among the most conspicuous constellations in the north- 
ern hemisphere are the Lesser Bear, in the direction of 
which the north pole of our earth continually points, the 
Great Bear, which is more distant from the pole, Perseus, 
Cassiopeia, Lyra, Hercules, the Wagoner, Orion chasing 
the Pleiades and Hyades, while the Dog Star, though in 
the southern hemisphere, follows in the train. The south- 
ern constellations are more brilliant. The Southern Cross, 
the Argo, the Southern Triangle, the Centaur, and the 
Southern Crown, are among the most splendid. 

A new system of arrangement and nomenclature has 
been proposed, in which the heavens should be covered by 
a net-work of imaginary circles crossing each other at 
regular intervals, so that each star could be referred to its 
exact place. Meanwhile much confusion must exist where 
3,487 stars are to be formed into 94 figures whose outlines 
are imaginary and undefined, neither coincide with the po- 
sition of the stars nor are definite in themselves, and seem 
made uncouth and perplexed purposely to baffle the ob- 
server. 

The zodiacal constellations which we have described are 
12 in number, and contain 1,016 stars. The northern 
constellations are 34 in number, and contain 1,444 stars. 
The southern constellations are 47 in number, and contain 
1,027 stars. A large number of these are telescopic stars, 
but they are well known. 

These constellations should be seen on a map or a con- 
cave celestial globe. Such a globe represents them as 
they really are, and a convex one reverses their appear- 
ance to us. 

§ 326. In a celestial map the eastern part of the 
heavens is toward the left hand, the western part toward 
the right. If we stand on the earth with our face toward 
the north, we have the eastern part of the earth and of 
the heavens on our right hand. But if we face these 
heavens, and make a map of them, the eastern part must 
be depicted on the left hand. The eastern part of the 



ELEMENTS OF ASTRONOMY. 237 

heavens is that you reach by travelling eastward, and 
when you face it it appears on the left hand. 

In all drawings of the celestial motions in this book, the 
west is to the right hand, the east to the left hand. On 
maps of the earth, the east and west points are differently 
placed. In looking at the heavens we look at a concave 
sphere, in looking at the earth we have a convex sphere, 
and hence the maps which represent portions of each are 
differently made. For the concave sphere the west is 
on the right, and the east on the left hand. For the 
convex sphere the east is on the right, the west on the left 
hand. 

We must remember that north and south are points, 
the imaginary ends of the pole of rotation, while east and 
west are directions. This difference exists both in the 
earth and the heavens. One definite part of the globe 
is called the northern point, but no point or part is called 
the eastern in this sense. If we travel north on the sur- 
face of the globe we come to this point, where we can 
no longer go north, but must turn and go south. But we 
can travel east till we return to our starting place, nay we 
might go round the globe again and again and yet always 
travel eastward. Thus in the heavens the earth moves 
always in an east direction. There is no part of the 
heavens which is called the eastern, none which is called 
the western part. 



238 ELEMENTS OF ASTRONOMY. 



CHAPTER XIV. 

LAWS OF SHAPE AND MOTION. 

Attraction of Gravitation. Effect of Gravitation on the figures of the Sun 
and Planets. The Figure of the Earth that of Equilibrium. Illustration 
of the effect of Rotation on a Fluid Mass. Laws of Gravity. Centre of 
Gravity. 

§ 328. In the preceding pages many individual facts 
have been stated. The form, relative masses, and orbits 
of the members of the solar system have been given. We 
would now study the laws which decide these forms, and 
govern these motions. Before doing this we must make 
ourselves familiar with a principle whose workings pervade 
all we yet know of the universe, a principle which influ- 
ences the form and motions of all matter devoid of life, — 
the principle of gravity. Of the nature of gravity we 
know nothing ; we call it an attraction because in obe- 
dience to it bodies approach one another. But though we 
are ignorant of its nature and of its mode of action, we 
know with certainty the results to explain which we sup- 
pose its existence, and we know that they take place in- 
variably. 

Every particle of matter, as far as we know, attracts 
every other particle. If the particles are in a fluid or 
gaseous state, and are hindered by no other force, they 
rush together, and take the form which satisfies their mu- 
tual attraction. In this way we account for the form of 
the celestial bodies. If the bodies are solid, so that their 
particles cannot move easily, they are drawn to one another 
without losing their form. It is in this way that we ex- 
plain the motions of the planets. 

We will first consider what form would result from the 
gravitation of free particles toward each other, and then 
see how motion would modify this form. We will after- 
ward inquire what motions gravity, acting in connection 
with a primitive impulse, would impress on these bodies. 



ELEMENTS OF ASTRONOMY. 239 

§ 329. The globular form of the sun and planets 
makes it probable that they were once in a fluid state. 
A fluid mass, sufficiently removed from other bodies, is al- 
ways brought to a globular form by the equal mutual at- 
traction of its particles. The fluid particles move over 
one another with great ease. They continue to move until 
the attractions are all balanced, and the centre of gravity 
is at the same distance from every point in the surface of 
the mass. The only form which answers to this condition 
is a sphere ; for in a sphere the centre of form is also the 
centre of gravity. The spherical form is accordingly as- 
sumed by drops of rain, mist, quicksilver, &c. 

If there were no motion in the heavens we should proba- 
bly find all the heavenly bodies perfect globes or spheres. 
But we find motion, and this of a kind to affect the form 
which a body in a fluid state would assume. When a fluid 
mass rotates, all the parts rotate in the same time. Those 
which form the axis of rotation may be considered as merely 
turning round on themselves slowly ; those further from 
the centre describe circles with more rapidity, and those 
which are farthest from the axis of rotation describe the 
largest circles, and have the greatest rapidity. By this 
greater swiftness of rotation in the equatorial parts, centri- 
fugal force is generated, the equatorial particles try to fly 
off, the equilibrium of the globe (as it would be at rest) 
is destroyed. A part of the attraction which kept the 
equatorial particles in their place is balanced by the cen- 
trifugal force ; the rest is insufficient to retain them in 
their place ; they recede from the centre, and other parti- 
cles rush in to supply their place. 

§ 330. But the form of the mass is changed ; the par- 
ticles near the equator are piled up, those near the poles 
are depressed, until it reaches the state of equilibrium for 
a rotating fluid mass ; but if the mass while fluid stops ro- 
tating it must return to a globular form. If it hardens 
while rotating, it will retain the spheroidal form. And its 
departure from the truly spherical form will be in propor- 
tion to the rapidity of rotation. "Very rapid rotation 
causing great centrifugal force raises the equatorial parti- 
cles and depresses the polar parts exceedingly. The dif- 



240 ELEMENTS OF ASTRONOMY. 

ferent planets have now varying rapidities of rotation ; and 
those which rotate most rapidly are most flattened at the 
poles. 

It is found by calculation that the rotation of the earth 
is precisely rapid enough to give it the flattening which it 
has. If it rotated more rapidly, the water at the equator 
would be flung off. Or if it were a larger body, and ro- 
tated in twenty-four hours as it now does, the equatorial 
particles would likewise be in danger. 

§ 331. The effect of rotation on the form of a fluid 
mass may be seen by spinning a pail partly full of water 
suspended by a string. The surface of the water instead 
of remaining horizontal will become concave. The centri- 
fugal force generates in all the water a tendency to leave 
the axis and to press toward the circumference. It is 
therefore urged against the pail, and forced up the sides, 
till the excess of height, and consequent increase of pres- 
sure downwards, just counterbalances its centrifugal force. 
If the rotation becomes very rapid, the surface of the 
water becomes more concave ; if allowed to diminish, it 
becomes less so. In a similar way more or less rapid rota- 
tion would increase or diminish the excess of the equatorial 
parts. 

The following pretty little experiment has been tried to 
show the globular form which a liquid relieved from all ex- 
ternal attraction or pressure would take, and also the 
spheroidal form which rotation would induce. 

Placing a mixture of water and alcohol in a glass box, 
and therein a small quantity of olive-oil, of density precisely 
equal to the mixture, we have in the latter a liquid mass 
relieved from the operation of gravity, and free to take the 
exterior form given by the forces which may act upon it. 
In point of fact, the oil instantly takes a globular form, by 
virtue of molecular attraction. A vertical axis being intro- 
duced through the box, with a small disc upon it, so ar- 
ranged that its centre is coincident with the centre of the 
globe of oil, we turn the axis at a slow rate, and thus set 
the oil sphere in rotation. " We then presently see the 
sphere flatten at its poles and swell out at its equator, 
and we thus realize, on a small scale, an effect which is 



ELEMENTS OF ASTRONOMY. 241 

admitted to have taken place in the planets. The spheri- 
fying forces are of different natures, that of molecular at- 
traction in the case of oil, and of universal attraction in 
that of the planet ; but the results are analagous, if not 
identical. Quickening the rotation makes the figure 
more oblately spheroidal. When it comes to be so quick 
as two or three turns in a second, the liquid sphere first 
takes rapidly its maximum of flattening, then becomes hol- 
low above and below, around the axis of rotation, stretch- 
ing out continually in a horizontal direction, and finally 
abandoning the disc, is transformed into a perfectly regu- 
lar ring. At first this remains connected with the disc- 
by a thin pellicle of oil ; but on the disc's being stopped, this 
breaks and disappears, and the ring becomes .completely- 
disengaged. 

§ 332. We have spoken as if the fluid mass of the 5 
earth were first formed into a globe, and afterward, from 
rotation, took on the spheroidal shape. Probably the mo- 
ment it was isolated it took its form, which rotation, begin- 
ning at the same moment, modified. The same force 
which isolated it, probably communicated by one impulse 
the rotary and onward motion. 

The fact that these two motions throughout the solar 
system almost invariably take place in the same direction, 
makes the probability that they were communicated by one 
impulse as millions to one. For simplicity, let us consider 
the case of the earth only, and study how these two mo- 
tions might have arisen. Whether the fluid matter which 
we have good evidence once composed the earth, was 
brought from some other part of space within the sphere 
of the sun's attraction, or whether it was thrown off the ro- 
tating sun, as splinters are sometimes cast off grindstones, or 
whether from some other cause it received an impulse, we 
cannot but suppose it to have had some inclination to move 
in some direction. Newton supposed that all the planets 
had received an impulse in a straight line in a tangent to 
their present orbits. The projectile force could not have 
passed through the centre of gravity, for the earth rotates. 
If the direction of the impulse was through the centre of 
the sphere, equal velocities would have been communicated 
21 



242 ELEMENTS OF ASTRONOMY. 

to all parts of the sphere, and no rotation would have 
taken place. But if the force was directed a little on one 
side of the centre of gravity, the equilibrium of the parti- 
cles would be destroyed. Those which received the blow 
would be carried down with great rapidity, and the other 
half of the sphere would rise. 

§333. The axis of rotation would immediately be 
formed in the line where all the opposing forces balance 
each other, and the particles which received the stroke 
would move round it, carrying with them all the particles 
in the same hemisphere with themselves. The place which 
received the impulse would become a point in the equator. 

But simultaneously with the formation of the axis of ro- 
tation the sun must have attracted to itself one half of 
the sphere, at right angles to the direction in which the pro- 
jectile force acted. A plane passing through the centre of 
the sun and the line representing the projectile force marks 
the plane of the ecliptic. If the sun were below the plane 
of the earth's equator, it would draw the earth down, and 
cause the ecliptic to cut the equator as it now does. It 
would make the poles of the orbit inclined as they are to 
the poles of the equator. 

Rotation once established, nothing could occur to change 
its poles, and we have every evidence that they have re- 
mained unaltered as far as man's records extend. 

The axis always remains parallel to itself however much 
it may be inclined to the axis of the onward motion. This 
may be shown by throwing into the air a homogeneous 
globe pierced with an axis, and impressing on it at the 
same time motion of rotation. Whether this axis is perpen- 
dicular to the curve described by the body or not, it will 
always remain parallel to itself. 

When the direction of the force does not pass near the 
centre of gravity, great velocity of rotation is induced. 
The part of a sphere in which the force has been applied 
may be found by calculation. In case of the earth it was 
in some part of the equator, passing about twenty-five miles 
from the centre. The remoter planets must have been im- 
pelled in a direction passing farther from the axis of revo- 
lution, for their rotation is extremely rapid. 



ELEMENTS OF ASTRONOMY. 243 

The rotary motion requires no expenditure of force. It 
merely results from the unequal application of the force. 
None of the force applied is used up by it, because the 
body as a whole is in a state of rest. As many of the par- 
ticles move backward as forward, and the centre of gravity 
undisturbed by rotation remains unmoved. 

§ 334. On the earth, where there is so much friction, 
a blow which is unable to overcome the inertia of a body 
and friction, will, if its direction passes through the centre, 
cause the body to shake ; if it passes on one side of the 
axis it may cause a slight rotary motion. We feel there- 
fore as if force were always consumed in causing rotation. 
But it is a fact that equal impulses will carry equal bodies 
over equal spaces in equal times whether the bodies rotate 
or not. And this is not only true of a rotating body, but, 
as we shall presently see, of a system of bodies revolving 
round a common centre. On the earth, when friction is re- 
moved as much as possible, a very slight force is sufficient 
to destroy equilibrium and cause motion. If a hundred 
pounds could be placed in the scale of a delicate balance, 
a weight scarcely more than sufficient to overcome the 
friction of the machine would cause one scale to descend 
and the other scale to ascend. In the heavens, where 
there is no friction to be overcome, the slightest inequality 
of the two forces acting on the balanced particles of a sphere 
is sufficient to make the whole sphere rotate. 

§ 335. The onward motion, or translation of a body 
always requires an expenditure of force, and equal forces 
"will carry the centre of gravity forward through equal 
spaces, in whatever part they may be applied. This is 
true of different parts of a solid body ; of two bodies con- 
nected by a pole ; or of two unconnected bodies. 

Suppose two bodies, weighing one pound each, and con- 
nected by a pole, to receive through their centre of 
gravity a blow which carried them forward two feet. An 
equal blow, not on the centre of gravity of the two bodies, 
but on one of them, would, if that were alone, carry it for- 
ward four feet ; but as it is fastened to the other body, the 
force is divided between the two, and therefore carries 
them and their common centre of gravity forward two feet ; 



244 ELEMENTS OF ASTRONOMY. 

producing also in the two bodies a whirling motion round 
their centre of gravity. If the bodies have no connection, 
and one receives a blow, it moves forward four feet, but 
the other is stationary ; the centre of gravity is half way 
between the two, and in this case also advances two feet. 

A cluster of stars may thus move onward unimpeded by 
their individual revolutions round their common centre of 
gravity, and the earth and moon make their little evolu- 
tions, while their common centre of gravity sweeps steadily 
round the sun. 

§ 336. Since the common direction of the two plan- 
etary motions points to a common origin, and since one im- 
pulse can account for both motions we need seek no other 
cause for either motion. Not only is it very probable that 
one cause produced both motions, but the chances are mil- 
lions against one that any cause should have produced one 
alone. So that if we find rotation or revolution alone we 
should be inclined to suppose that both motions had ex- 
isted, and that one had been stopped by some external 
force. To cause revolution alone, we have seen that the 
direction of the force must be through the centre, which is 
extremely improbable. Rotation alone can be caused on 
earth when onward motion is destroyed by friction, but in 
the heavens, where there is no resistance, any force whose 
inequalities could cause rotation, must itself cause revo- 
lution. Wherever there is revolution, therefore, rotation 
is probable, as in the case of the planets most distant from 
the sun, which have not been observed. Wherever there 
is rotation we may infer translation in space, even when we 
cannot perceive it, as in the case of the sun. Or we may 
suppose that a force acting through the centre of gravity 
in the opposite direction, has put an end to the onward 
movement without interfering with rotation. The two mo- 
tions are independent of one another, and one may outlast 
the other. Thus a top often twirls on the same spot after 
it has ceased to describe circles. 

The period of a planet's rotation depends on the place 
in which it is struck, and the force of the blow compared 
to its mass. 

The period of a planet's revolution depends on the mass 



ELEMENTS OF ASTRONOMY. 245 

and distance of the attracting body and the force of its 
projection into space. 

§ 337. In consequence of rotation every particle on 
the surface of the earth describes a circle once in twenty- 
four hours. In consequence of revolution, the whole earth, 
and of course each place on its surface, describes, in the 
course of 365 days, an ellipse, of which the sun is in one 
of the foci. We have learned the influence of gravity on 
the shape of bodies, we will now seek the effect of gravity 
on a system of bodies, in establishing a common centre, 
and consider the motions which gravity and a primitive 
impulse would impress on bodies. 

The simple action of gravity can only be seen in the 
heavens, in the attraction one heavenly body exerts on 
another. When bodies on the surface of the earth are 
drawn to the earth, this drawing, which we call their 
weight, is not the full measure of gravity. Other forces 
are at work. The whirling of the earth gives all bodies 
in inhabitable parts of its surface a tendency to fly off, 
which partly counteracts the force of gravity. The gravi- 
tation of a body to the earth is then the force with which 
gravity draws it, minus its centrifugal force. Even at the 
equator, however, the difference between the power of 
gravity and the actual gravitation is trifling ; it is but 28~¥ 
of the whole weight. As the tendency to fly off can be 
calculated, the true amount of gravity at the earth's sur- 
face can be ascertained. At the poles the gravitation of 
bodies is an exact measure of the force of gravity residing 
in a body of the mass of the earth, and exercised on a body 
removed from the centre of gravity by a distance equal to 
the polar radius. 

§ 338. The first law of gravitation is this. The at- 
traction of one body on another does not depend on the 
mass of the body attracted, but is the same whatever be 
the mass, if the distances are the same. 

Thus Jupiter attracts the sun, and Jupiter attracts the 
earth also ; but though the sun's mass is three hundred 
thousand times as great as the earth's, yet the attraction 
of Jupiter on the sun is exactly equal to his attraction on 
the earth, when the sun and the earth are equally distant 
21* 



246 ELEMENTS OF ASTRONOMY. 

from Jupiter. When the sun and earth are at equal dis- 
tances from Jupiter, the attraction of Jupiter on the sun 
draws it through as many inches or parts of an inch, in one 
second of time, as it draws the earth in the same time. 

The second law of gravitation is this. Attraction is 
proportional to the mass of the body which attracts, if the 
distances of different attracting bodies are the same. 

Thus suppose the sun and Jupiter are at equal distances 
from Saturn. The sun is about 1,000 times as large as 
Jupiter. Then whatever be the number of inches through 
which Jupiter draws Saturn in one second of time, the sun 
draws Saturn in the same time through 1,000 times that 
number of inches. 

The third law is this. If the same attracting body acts 
upon several bodies at different distances, the attractions are 
inversely proportional to the squares of the distances from 
the attracting body. Thus the earth attracts the sun, and 
the earth also attracts the moon. But the sun is 400 
times as far off as the moon, and therefore the earth's at- 
traction on the sun is only ye <jWtt P ar ^ of its attraction on 
the moon. Or as the earth's attraction draws the moon 
through about ^V of an inch in one second of time, the 
earth's attraction draws the sun through ^fct^u^o- of an 
inch in one second of time. 

§ 339. The reader may ask, " How is all this known 
to be true ?" The best answer is perhaps the following. 
We find that the force which the earth exerts upon the 
moon bears the same proportion to gravity on the earth's 
surface which it ought to bear in conformity with the rule 
just given. For the motions of the planets, calculations 
are made which are founded upon themselves, and which 
will enable us to predict their places with considerable ac- 
curacy if the laws are true, but which would be much in 
error if the laws were false. The accuracy of astronomical 
observations is carried to a degree which can scarcely be 
imagined. And by means of these we can every day 
compare the observed place of a planet with the place 
which was calculated beforehand, according to the law of 
gravitation. It is found that they agree so nearly as to 
leave no doubt of the truth of the law. The motion of 



ELEMENTS OF ASTRONOMY. 247 

Jupiter, for instance, is so perfectly calculated, that as- 
tronomers have computed ten years beforehand the time at 
which it will pass the meridian of different places, and the 
predicted time is found to be correct within half a second 
of time. 

§ 340. Since all the planetary bodies are of a spher- 
oidal form, the labors of astronomers are much shortened 
by considering all lines of attraction as passing from one 
centre to another. In a sphere the centre of form is also 
the centre of gravity, that is, it is the point round which 
the weight of the body is equally distributed in all direc- 
tions. 

It is often necessary to ascertain the position of the 
common centre of gravity of two or more bodies connected 
together ; and this is not difficult when we know the cen- 
tres of gravity of the several bodies themselves. (Fig. 6, 
Plate I.) Let A and B be two globes connected by a 
rod, which we shall suppose, for the sake of simplifying 
the explanation, to have no weight in itself. The centres 
of these globes are their centres of gravity. And as we 
may regard all their weight as acting from those points, 
the same reasoning which enabled us to understand that 
the forces acting upon the different parts of any one body 
may balance each other round a certain point, leads to the 
belief that a point may exist in which we may regard the 
actions of A and B as jointly and equally exercised. This 
point is evidently somewhere in the line A B, which joins 
their centres. It is determined on the principle of the 
lever, by dividing the line A B into two such parts that 
the distance of each body from the point C shall be pro- 
portional to the weight of the other. Thus suppose A to 
weigh 6 lbs., and B 1 lb., then A's distance from C must 
be to B's distance as one to six. A support placed at C 
will sustain them both at rest, and will be pressed upon 
with the weight of both combined. But owing to the 
greater distance of B from C, and its inferior size, a given 
force applied to it will carry it through six times the space 
through which the same force would carry A. If both 
bodies therefore were moving round their common centre 
of gravity, they would perform their orbits in the same 



248 ELEMENTS OF ASTRONOMY. 

time, but B would move with six times the velocity 
of A. 

§ 341. Supposing that a third body, D, were con- 
nected with A and B, by a rod proceeding from the point 
C. Then the common centre of gravity of all three bodies 
will be in the line C D, since the weights of A and B may 
be regarded as concentrated at C and act as if a single 
body of their total weight were placed there ; and it may 
be determined in the same manner as before. In like 
manner, if another body, F, be connected with the system, 
by a rod uniting it with the rest at their common centre of 
gravity, E, the centre of gravity of the four will be in the 
line between E and F. And it will be at such a distance 
from F, that its weight multiplied by the distance F G, 
shall be equal to the combined weights of the other bodies 
acting at the distance G E. In this manner any number 
of bodies may be connected with the system ; or in a sys- 
tem already existing, we may ascertain the common centre 
of gravity by a similar process. 

As the rod connects the globes so does gravity hold the 
celestial bodies in their places. But since all, at least in 
our system, are in motion, the balance must be each mo- 
ment struck anew. Each orb is exposed to the influence 
of every other orb, and by their incessant deviations from 
regular motion, the equilibrium and stability of the whole 
are preserved. When Jupiter passes on the same side of 
the sun with the earth, the earth cannot but feel his at- 
traction, and Jupiter and all his moons recognize the ap- 
proach of the little earth. A balance is struck between 
the sun's attraction and that which Jupiter in his present 
place exercises on the earth. The earth moves toward 
Jupiter till the equilibrium is restored and modifies her path 
by his influence. All this is not done by jerks or intervals. 
Gravity acts incessantly, restoring order as rapidly as it is 
disturbed. Every change is effected gently, swiftly, and 
so far as we can judge noiselessly. 

§ 342. Let us suppose two bodies newly suspended in 
space ; both bodies would rush together and meet at one 
point. If one body were heavier than the other, the point 
of meeting would be proportionally near the heavier body. 



ELEMENTS OE ASTRONOMY. 249 

If they were equal in weight, they would meet mid-way. 
Now let us suppose the second body to have an onward 
motion given to it. The moment it felt the attraction of 
the first body, it would attract that in return, share its mo- 
tion with it, and force it to move round their common centre 
of gravity. If the two bodies were of equal weight, they 
would revolve at equal distances from the centre of grav- 
ity ; if not, the heaviest would move in the smallest orbit ; 
but both would revolve in the same time. 



CHAPTER XV. 

LAWS OE MOTION — (CONTINUED.) 

Three general Laws of Motion. Composition of Forces. Path of a Pro- 
jectile near the Earth's Surface. Motion in a Curve. Projectile and 
Centripetal Forces. Motion in the Solar System. Kepler's Laws. 
Central Forces. 

§ 343. There are three general laws which a body 
obeys in its motion, whatever be the kind of body or the 
kind of force that impels it ; whether it be a particle of 
dust driven by the wind, or a planet revolving in conse- 
quence of an original impulse through the celestial spaces. 

A body does not change its state, either of rest or mo- 
tion, unless in consequence of some external cause. 

The effect is always proportional to the force impressed, 
and takes place in the direction in which the force acts. 

Action and reaction are equal and contrary. This law 
holds whether the bodies attract or repel one another, and 
whether they act at a distance or in apparent contact. 

A body is often acted on by several forces at once, and 
the effect of their joint action is an exact compound of their 
several effects, or the same as if each had acted succes- 
sively. 

The body may be acted on by two forces applied at the 
same point of the body. If they act in the same direction, 



250 ELEMENTS OF ASTRONOMY. 

the resultant will be in that direction, and equal to their 
sum. If in opposite directions, the resultant will be in 
the direction of the greater, and equal to their difference. 
If at an angle, the resultant will be in the same plane, and 
represented by the diagonal of a parallelogram of which 
the two sides represent the simple forces. In this case it 
is less than the sum of the forces, and greater than their 
difference. 

Thus a body may move in a certain direction in conse- 
quence of one, two, or a dozen impulses. 

It is often desirable to resolve a single force into others 
to which it is equivalent, in order to find its effect in a 
given direction. We need only resolve the force into two, 
one of which is parallel to the given direction, and the 
other at right angles to it. The latter can have no effect 
in the given direction, and therefore the other will express 
the whole effect. 

Thus when we wish to know how much one planet draws 
another from the plane of its orbit, the line representing 
the influence of the planet and in the direction of their 
centres is considered the diagonal of a parallelogram of 
which one side shows how much the planet is moved from 
its orbit. 

Celestial motions are caused, not by two impulses, but 
by an impulse and a pressure. The impulse was imparted 
by a force of whose nature we can only form a vague guess. 
The pressure is the constant attraction of gravitation. If 
the heavenly bodies had received only the impulse, they 
would have moved on in straight lines forever. If other 
impulses had interfered successively, they would have moved 
on in broken lines, making angles with each other. Since 
the second force is a pressure, and acts incessantly, the 
straight lines will become infinitely short, and the path 
consequently will become curvilineal. Since a body, if 
left to itself, moves in a straight line, we may conclude, 
when it moves in a curve, without being compelled to it by 
a fixed obstacle, that there is a force of pressure constantly 
deflecting it from the direction of the tangent. We are 
now therefore ready to consider the effect of attraction on 
the motion of bodies. 



ELEMENTS OF ASTRONOMY. 251 

§ 344. If a body is projected in the direction in which 
gravity draws it, its velocity is increased. If gravity acts 
directly contrary to the projectile force, it gradually 
weakens and at length overcomes it, as when an arrow shot 
vertically is brought to the ground. A more important 
case for astronomy than either of these is when the body 
is projected transverse to the direction in which the force 
draws it. 

The simplest instance of this motion that we can imagine 
is the motion of a stone when it is thrown from the hand 
in a horizontal direction. It does not move in a straight 
line. It begins to move in the direction in which it is 
thrown ; but this direction is speedily changed. It con- 
tinues to change gradually and constantly, and the stone 
strikes the ground moving at that time in a direction much 
inclined to the original direction. The most powerful 
effort that we can make is not sufficient to prevent the 
body from falling at last. This experiment therefore will 
not enable us to judge immediately what will become of a 
body (as a planet) which is put in motion at a great dis- 
tance from another body which attracts it (as the sun). 
But it will assist us much in judging generally what is the 
nature of the motion when a body is projected in a direc- 
tion transverse to the direction in which the force acts 
upon it. 

§ 345. The general nature of the motion is this. The 
body describes a curved path, of which the first part has 
the same direction as the line in which it is projected. 

If A (Fig. 8, Plate I.) is the point from which the stone 
was thrown, and A H the direction in which it was thrown ; 
and if we wish to know where the stone will be at the end 
of any particular time, (suppose three seconds,) and if the 
velocity with which it is thrown would, in three seconds, 
have carried it from P to F, supposing gravity not to have 
acted upon it ; and if gravity would have made it fall 
from A to P, supposing it to have been merely dropped 
from the hand ; then, at the end of three seconds, the 
stone really will be at the point F. And it will have 
reached it by a curved path A F, of which different points 



252 ELEMENTS OF ASTRONOMY. 

can be determined in the same way for different instants of 
time. 

The calculation of the stone's course is easy, because 
during the whole motion of the stone gravity is acting 
upon it, with the same force and in the same direction. 
The motion of a body attracted by a planet or the sun, 
where the force varies as the distance alters, and is not 
the same either in amount or direction at the point F as it 
is at the point A, cannot be computed by the same simple 
method. But the same method will apply, provided we 
restrict the intervals for which the calculations are made 
to times so short that the alterations in the amount of the 
force and in its direction, during each of those times will 
be very small. Thus, in the motion of the earth, as af- 
fected by the attraction of the sun, if we used the process 
that we have described to find where the earth will be at 
the end of a month from the present time, the place that 
we should find would be very far wrong. If we calculated 
for the end of a week, since the direction of the force and 
its magnitude would have been less altered, the error would 
be much less than before. 

Fig. 8, Plate I., shows the paths described in obe- 
dience to several different combinations of the projectile 
force with gravity. 

§ 346. Every body which is under the influence of a 
constant attractive force, and of an impulse originally 
given, moves in a curved path round the centre of attrac- 
tion. 

And on the other hand, when a body moves in a curve, 
there must be one impulsive and one constantly restraining 
force acting upon it. If the projectile force acted at per- 
ceptible intervals, the body would describe the perimeter 
of a polygon, having as many sides as the number of im- 
pulses given. Since the projectile force acts at intervals 
infinitely small, the polygon has an infinite number of sides, 
is a circle. The projectile force urging the body on at a 
tangent, diminishes its tendency to the centre or the cen- 
tripetal force, and creates a centrifugal force in the oppo- 
site direction from the centripetal force. The centrifugal 
force arises from and is inseparable from a curvilinear 



ELEMENTS OF ASTRONOMY. 253 

track. It is not a tendency which the body originally has 
to fly from the centre, but arises from its constrained con- 
tinuance in a curved orbit, when, if unattracted to the 
centre, it would proceed in a tangent. 

The projectile and centripetal forces cannot be directly 
compared, for they are different in kind, one is impulsive 
and the other is incessant ; but the centrifugal force, gen- 
erated by a given projectile force, acts incessantly like the 
centripetal, and may be compared with it. We may also 
calculate through how long a space gravity must act to 
balance a given projectile force. 

§ 347. Every body moving under the influence of 
gravity describes one of the conic sections. 

If a cone is cut parallel to its base the section is a circle ; 
if cut obliquely to its base, but not in such a way as to in- 
tersect it, the section will be an ellipse more or less elon- 
gated. If it is cut parallel to the curved surface of the 
cone, the section will be a parabola ; if perpendicular to the 
base and not through the axis of the cone, it will be a hy- 
perbola. A cone divided by these sections best explains 
this. The curve described cannot be a circle unless the 
line of projection is perpendicular to the line of attraction, 
and unless the velocity with which the planet is projected 
is neither greater nor less than one particular velocity de- 
termined by the distance and mass of the attracting body. 
If it exceeds this velocity a little, or falls a little short of 
it, the body will move in an ellipse. 

If the projectile force gives a rapidity equal to that ac- 
quired by a body in falling through a height equal to one 
third the radius of the circle, and if also the projectile act 
at right angles to the centripetal force, the body will de- 
scribe a circle. 

If the projectile force is to that required for a circle as 
s/ 2 to 1, or equal to that acquired by a body falling 
through one half of the radius, the body will describe a 
parabola. 

Any ratio of the central forces between these two will 
cause the body to describe an ellipse. 

If the projectile force is stronger than that required for 
a parabola, or such as a body would gain by falling through 
22 



254 ELEMENTS OF ASTRONOMY. 

a height yet greater than one half of the radius, the orbit 
will be hyperbolical. 

§ 348. As it is extremely improbable that the two 
forces should be to one another in the definite proportions 
required to cause a circle or a parabola, we cannot expect 
to find these orbits in the heavens. 

Since ellipses and hyperbolas require no definite ratio 
of the forces, but vary in their axes according to the 
ratio, we may expect to find many orbits of these forms. 
Indeed we have no certainty that any except elliptic orbits 
exist in the heavens. The comets which have yet been 
recorded but once, may thousands of years hence manifest 
their elliptic orbits by revisiting us again. Meanwhile it 
requires a nice observation and calculation, to ascertain 
from the small portion of their orbit visible to us, whether 
it most resembles an ellipse , parabola, or hyperbola. 

If the projectile acts obliquely to the attracting force, 
and the velocity of projection is small, the body will move 
in an ellipse. If the velocity is great, it may move in a 
parabola or a hyperbola, but not in a circle. For even if 
the velocity of projection were just sufficient to make the 
body move in a circle, yet as it acts obliquely to gravity, 
gravity must either diminish or increase it, and thus pre- 
vent the body's moving in a circle. 

The earth when nearest the sun has a velocity of about 
102,300 feet a second, this being the result of its own 
projectile force, arising from the impulse which first set it 
in motion, and the power of the sun's attraction. By rea- 
son of this velocity it is constrained to move in an elliptic 
orbit. But if by any augmentation of the projectile force, 
the earth's velocity at this point were to amount to 144,700 
feet, the orbit would become parabolic ; and any velocity 
surpassing this would make its course hyperbolic. If its 
velocity were about 101,000 feet a second, or a little less 
than it now is, the orbit would be exactly circular. 

Thus various forms of orbits are produced by different 
projectile forces combined with one given attractive force. 
As the projectile force increases the body departs more 
and more from a circle. 

If a body describes a circle, the attracting body is in the 
centre of the circle. 



ELEMENTS OF ASTRONOMY. 255 

If it describes an ellipse, the attracting body is not in 
the centre of the ellipse, but in one focus. 

If it describes a parabola or hyperbola, the attracting 
body is in the focus. 

§ 349. The student should familiarize himself with the 
following terms which occur continually in speaking of 
ellipses. 

A straight line drawn from any point of the curve to the 
centre of attraction is called the radius vector. 

The angular velocity at any point of the curve is the 
velocity with which the radius vector at that point describes 
an angle. The actual velocity is equal to the space passed 
over divided by the time. Two bodies revolving round a 
common centre may have the same angular velocity ; but 
if one be twice as far from the centre as the other, its 
actual velocity will be twice as great. 

When the line which the body describes returns into 
itself, like a circle or an oval, it is called an orbit, and the 
time of describing the whole is called the periodic time. 

Fig. 9, Plate I. In the ellipse AEBD, S and H 
are the foci. Let S be that focus which is the place of 
the sun, if we are speaking of a planet's orbit, or the place 
of the planet, if we are speaking of a satellite's orbit. 

A B is the major axis of the ellipse. 

A C or C B is the semi-major axis. This is equal in 
length to S D. It is sometimes called the mean distance, 
because it is half way between A S, which is the planet's 
smallest distance from S, and B S, which is the planet's 
greatest distance from S. 

D E is the minor axis ; D C or C E the semi-minor axis. 

A is called the perihelion, B the aphelion of the orbit 
of a planet. In the moon's orbit they are called the peri- 
gee and apogee. 

A and B are called the apsides, and the major axis the 
line of apsides. 

The proportion which S C bears to A C is called the 
excentricity of the orbit. 

§ 350. If we know the mass of the central body, and 
if we suppose the revolving body to be projected at a cer- 
tain place in a known direction with a given velocity, the 



256 ELEMENTS OF ASTRONOMY. 

length of the axis major the eccentricity, the position of 
the line of apsides, and the periodic time may all be calcu- 
lated. 

In all our diagrams it is to be understood, that the 
planet, or satellite, moves through its orbit in the direction 
opposite to the motion of the hands of a watch. This is 
the direction in which all the planets and satellites would 
appear to move, if viewed from any place on the north side 
of the planes of their orbits. 

The deflection of a stone, thrown from the hand, from 
the straight line in which it began to move, exactly equals 
the space through which gravity would have made it fall in 
the same time from a state of rest, whatever may be the ve- 
locity with which it is thrown. Consequently when the stone 
is thrown with very great velocity, it will go a great dis- 
tance before it is much deflected from the straight line, 
and therefore its path will be very little curved. 

The same thing is true with regard to the motion of a 
planet. And thus the curvature of any part of the orbit 
which a planet describes will not depend simply on the 
force of the sun's attraction, but also on the velocity with 
which the planet is moving. The nearer the planet is to 
the sun, the more it will be drawn in, and its orbit curved ; 
but at the same time the greater velocity of the planet at 
any point of its orbit will tend to diminish the curvature of 
the orbit at that part. The same absolute curvature may 
be produced, as at the upper and the kwer end of an 
ellipse, by two more or less powerful centripetal and cen- 
trifugal forces. 

§ 351. Let us follow the planet through every part of 
its orbit. Suppose it projected at B with the necessary 
velocity. This velocity must not be so great that the at- 
traction of the sun will not bend its path very much. From 
B to D and A the attraction exceeds the centrifugal force, 
and velocity towards the centre is created. The sun's at- 
traction increases this velocity as the body moves towards 
A. The sun's attractive force also, on account of the 
planet's nearness, is very much increased at A, and tends 
to make the orbit more curved ; but the velocity is so much 
increased that the orbit is not more curved than at B. At 



ELEMENTS OF ASTEONOMY. 257 

A the attractive and centrifugal forces are equal, and from 
A to E and B the centrifugal exceeds the attractive, so 
that between A and B the velocity towards the centre is 
destroyed. After B the attraction is again in excess, and 
between that and A the velocity to the centre is increased. 
The sun's attraction retards it as the force of gravity re- 
tards a ball which is rolled up hill. "When it has reached 
A its velocity is comparatively small ; and therefore, 
though the sun's attraction at A is small, yet the deflec- 
tion which it produces in the planet's motion is, on account 
of the planet's slowness there, sufficient to make its path 
very much curved, and the planet approaches the sun, and 
goes on the same orbit as before. 

§ 352. The projectile force is that with which the cir- 
culating body would run off in a tangent to its path, if 
there were no centripetal force to prevent it. The projec- 
tile force at the first moment the body begins to describe a 
curve depends on the strength of the initial impulse. 
But after gravity has begun to act, the projectile force is 
increased or diminished according as gravity acts in or 
against the direction in which the body is moving. It 
must be remembered that it is the tendency which the body 
has to fly off in a tangent, and the strength of this ten- 
dency depends on the velocity of the body at each moment. 
This velocity may be greater or may be less than the in- 
itial velocity. When the body describes an ellipse, the 
projectile force is alternately greater and less than the in- 
itial velocity. When it describes a circle the velocity and 
the projectile force are the same. When a body is made 
to describe a circle, by moving along the concave surface 
of a sphere or cylinder, however much the velocity may 
increase the centrifugal force, the reaction of the surface 
will be increased in the same proportion, so that the body 
may describe the same circle with different degrees of 
velocity. 

But if the body is moving through space round a centre, 
and a great velocity be given to it, the centrifugal force 
will exceed the centripetal. Therefore the body will be 
driven to a greater distance than before, from the centre, 
and will describe a curve exterior to the circle. For a 
22* 



258 ELEMENTS OE ASTRONOMY. 

like reason, if the velocity be diminished, the centrifugal 
force becoming less than the centripetal, the body will de- 
scribe a curve interior to the circle. 

In general, the centrifugal force is directly as the square 
of the velocity of the revolving body, and inversely as 
the radius of curvature of the arc which it describes. 

§ 353. Three important laws of planetary motion 
were discovered by Kepler, and pass under his name. 

I. The orbits of all the planets are ellipses, of which 
the sun occupies one focus. 

II. The radius vector of the planet describes equal 
areas in equal times. 

III. The squares of the periodic times are as the 
cubes of the mean distances of the planets, or as the cubes 
of the major axes of their orbits. 

The fact of the ellipticity of the orbits he determined 
from direct observation of Mars when in different parts of 
his orbit. Finding the distances of Mars incompatible with 
the supposition that he moved in a circle, he tried them 
with an ellipse, and found they corresponded. Upon trial 
of the other orbits, he found them also elliptic. 

The variations in the apparent diameter of the sun like- 
wise agree with the supposition that the orbit is elliptic. 

If we draw an ellipse, and from one focus, the position 
of- the sun, draw radii to the positions of the earth at equal 
intervals, thus dividing the surface of the ellipse into 
sections, these sections will be equivalent. Those which 
rest on short arcs are included within longer sides, and 
those which have long arcs are included within shorter 
sides. Thus the radius vector describes equal areas in 
equal times. 

Having discovered the relative mean distances of the 
planets from the sun, and knowing their periodic times, 
Kepler endeavored to find if there was any relation be- 
tween them, and thus discovered his third law. 

§ 354. The following are the most important proposi- 
tions concerning the motion of bodies acted on by central 
forces. 

If the centre of attraction remains always in the same 
place, the curve will be wholly in one plane, passing 



ELEMENTS OF ASTRONOMY* 259 

through that centre ; and the areas described by the radius 
vector will be proportional to the times of description. 

First. Suppose the central force to act during equal 
finite intervals of time ; suppose C (Fig. 1, Plate II.) the 
centre of attraction, A B the line passed over in one of the 
equal intervals, the body with its uniform motion would, 
during the next equal portion of time, go over a line B D 
=A B, but at B it is acted on by the central force. Sup- 
pose the momentary action is such, that in the same time 
the body would move along B E, then completing the par- 
allelogram, B F will be the real line of motion. Joining 
CD, the triangle ABC— triangle B C F, because they 
have equal bases, and the same altitude. The same thing 
may be shown with regard to the next triangle, &c. 
Hence, the sum of all the triangles, or the whole area de- 
scribed in a given time, will be proportional to the time of 
description. 

As the force acting on the body is supposed to be in the 
direction of the plane of ABC, it has no tendency to 
move it out of that plane, and therefore B C F is in the 
same plane with ABC. The same is true of the next 
triangle, and of the whole area described. 

§ 355. If the curve described lies wholly in one plane, 
and the radius vector, drawn from a certain point in the 
plane, always describes around that point areas proportional 
to the times, that point is the centre of attraction. 

For around any other point than the centre of attrac- 
tion, the areas described in equal times cannot be equal. 
Thus, (Fig. 1, Plate II.,) take any point, G, it is evident 
that G B F cannot be equal to G B A ; for then D F would 
need to be parallel to G B, whereas it is parallel to C B. 

§ 356. The projectile velocity at any point of the 
curve is inversely as the perpendicular let fall on the tan- 
gent at that point from the centre of attraction. 

For the small triangle described in an instant, by the 
radius vector being every where of the same area, its base 
must be inversely as its perpendicular ; but the base is the 
projectile velocity, and the perpendicular on the base is the 
perpendicular on the tangent. 



260 ELEMENTS OF ASTRONOMY. 

§ 357. If there be two free bodies, the one cannot 
remain at rest, while, by its attraction, it causes the other 
to move round it ; but if the two bodies receive equal im- 
pulses in opposite and parallel directions, their centre of 
gravity will remain at rest, and they will describe similar 
curves. The first part of the proposition is manifest, for 
as one body attracts the other, the other will attract the 
first, and cause it to approach. 

Let A and B (Fig. 2, Plate II.) be the two bodies, C 
their centre of gravity. It follows, that A C : C B : : B : A. 
Let the bodies receive equal and parallel impulses in the 
directions B F and A G, and suppose that, in a given in- 
terval, the body A would move along A G ; join G C, and 
produce it to F ; B F will be the line passed over by B in 
the same moment ; for the impulses being equal, the veloci- 
ties will be inversely as the masses, that is, directly as 
A C : B C ; but by similar triangles A G : B F: : A C : B C. 
Again, suppose that, in consequence of the mutual attrac- 
tion, the bodies describe the curves B D, A E, then G E, 
F D will be the momentary deflections. G E will be to 
FD : : B : A; GC: CF: : AC : CB: :B: A; and 
hence the remainder E C will be to the remainder C D 
also in the same proportion, viz : — : : B : A. Hence the 
same point C will still be the centre of gravity of the two 
bodies when they have arrived at E and D, and will be so 
continually. 

Again, the small arcs A E and B D are similar, since all 
the straight lines connected with the one are proportional 
to the corresponding lines connected with the other. The 
arcs described the next moment will be similar for a like 
reason ; and hence the whole arcs described in equal finite 
times will be similar, and the whole curves described by A 
and B will be similar. 

§ 358. The angular velocity at the centre of force is 
inversely as the square of the radius vector. For the area 
of the indefinitely small triangle ABC (Fig. 1, Plate II.) 
is expressed by B C 2 X angle A C B. But in equal inter- 
vals of time, these areas are equal in all parts of the orbit. 
The angle A C B must therefore be inversely proportional 
to B C 2 , or to the square of the radius vector. 



ELEMENTS OP ASTRONOMY. 261 

§ 359. To determine the ratio of forces by which 

bodies tending to the centres of given circles are made to 

revolve in their peripheries. Let A M a (Fig. 3, Plate II.) 

be the circle in which one of the bodies moves round the 

centre of force E, and let the indefinitely small arch A 

be the distance it moves over in a given small interval of 

time. The centripetal force will be proportioned to A p. 

The chord and arc A will be equal in length. Whence, 

A 2 =AaxAp=ACx2 ap ; consequently, 2 Ap= 

A O 2 

-r—^- . And the same may be shown with respect to motion 

in any other circle. So that if R and r denote the radii of 

two circles, F and f the respective central forces, V and v 

the velocities with which the bodies move in their periphe- 

V 2 v 2 
ries, we shall have F : f : : -=- : — ; therefore the forces 

E, r 

are as the squares of the velocities directly, and as the 

radii inversely. 

y2 v 2 
Cor. 1st. Because F : fV: — : — it follows that 

R r , 

Y : v : : y RF : yr f, and 

t? V 2 v 2 

R:r:: -Y'-T. 

§ 360. The centrifugal force may now be compared 

with gravity, for if v be the velocity of a particle moving 

in the circumference of a circle of which r is the radius, its 

v 2 
centrifugal force is f= — . Let g be the constant 

force of gravity, and h the space or height through 
which a body must fall in order to acquire a ve- 
locity equal to v; then v 2 =2hg. for the accelerating 

force in the present case is gravity ; hence f = . If 

we suppose h=Jr, the centrifugal force becomes equal to 
gravity. 

Thus, if a heavy body be attached to one extremity of a 
thread, and if it be made to revolve in a horizontal plane 
round the other extremity of the thread fixed to a point in 
the plane ; if the velocity of revolution be equal to what 



262 ELEMENTS OF ASTRONOMY. 

the body would acquire by falling through a space equal to 
half the length of the thread, the body will stretch the 
thread with the same force as if it hung vertically. 

§ 36 1 . Since the versed sine of an arc of a circle is 
equal to the square of the corresponding chord divided by 
the diameter, and the chord of a very small arc nearly 
equals this arc, the square of this arc divided by the diam- 
eter, gives the versed sine. Let r be the radius, t the 
ratio of the circumference of a circle to its diameter, and 
T the time of revolution expressed in seconds. The arc 

actually described in a minute of time == -7^— . This squared 

and divided by the diameter gives, for the versed sine, 
which is proportional to the attracting force, the expression 

-rf^- . But the attracting forces are in inverse proportion 

to the squares of the radii. Hence we have 

-11 ,/• -;. 2 r?r 2 . 2r'x 2 

r 2 * r /2 ' * ^2 * »JV2 
r 3 ^2-— j./3 rj\ 2 . 

which is Kepler's third law, that the squares of the pe- 
riodic times are proportional to the cubes of the mean 
distances. 

This same proposition gives us the mass of two attracting 
bodies, the orbits and periodic times of two bodies revolving 
round them being known. The forces are the masses, 
and may be found by dividing the cube of the distance 
of a body moving round one of them divided by the 
square of its time, with the cube of the distance of the 
other divided by the square of its periodic time. Thus the 
cube of the moon's distance divided by the square of its 
periodic time is to the cube of the earth's distance divided 
by the square of her periodic time (nearly) as the mass of 
the earth is to that of the sun. 

§ 362. By means of the preceding propositions sev- 
eral practical questions of interest are solved. The dis- 
tance of a revolving body from the earth's centre being 
known, its velocity and periodic time may be deduced. 
Thus, let the radius of the earth (=21,000,000 feet, 



ELEMENTS OF ASTRONOMY. 263 

nearly,) he denoted by r, and the space through which a 
heavy body falls in on© second at the surface (=16J feet), 
by £g, the force of gravity at the surface being denoted 
by g ; then will the velocity per second in a circle at the 
surface be=^/2 gr:=26,000 feet nearly ; and the time 
of revolution =5,075 seconds. Let R be put for the 
radius of any other circle described by a projectile about 
the earth's centre : then, because the force of gravi- 
tation about the surface varies inversely as the square of 

the distance, we have, — - : —To ' : 26,000 feet (veloc- 
ity per second at the surface) : 26,000 jsJ ■= , the velocity 

in the circle whose radius is R. And r 2 : R 2 : : 5,075s 

R 3 
(the periodic time at the surface) : 5,075 y"— — , the pe- 
riodic time in the circle whose radius is R. 

For example, if R be assumed equal to 60 r, the dis- 
tance of the moon from the earth, the expression for the 
velocity will become 3, 356 J feet per second ; and that for 
the periodic time will become 2,360,035s. or 27t<j days, 
nearly. 

§ 363. Or, knowing the periodic time of the moon, 
and the radius of its orbit (240,000 miles), we can calcu- 
late the space through which she would fall if left to her- 
self, in a minute. This space will be the versed sine of 
the arc described in that time. The arc is easily found ; 
for as the moon takes 27 days, 7 hours, 43 minutes, to de- 
scribe her whole orbit, the following proportion will give it. 
As 27, 7', 43" : 1" : : 360° : 33" nearly ; of this arc the 
versed sine may be computed ; it is one half of a tenth of 
an inch, taking the moon's distance to be 240,000 miles. 

Now if the force which retains the moon in her orbit is 
identical with terrestrial gravity, it must decrease as 
the square of the distance. That is, if the moon is 60 
times as far from the earth's centre as a body near the sur- 
face of the earth, the space described by the moon should 
be to that described by a falling body near the earth's sur- 
face as 60 2 : l 2 ; the time being the same. A body falls 



264 ELEMENTS OF ASTRONOMY. 

through 59,400 feet in one minute near the earth's surface ; 
hence, as 60 2 : l 2 ; or as 3,600 : 1 : : 59,400 feet : 
16 J feet, which is the space through which a body would 
fall in a minute at the distance of the moon. Now 
this agrees, making allowance for using round numbers, 
with the actual distance through which the moon would 
fall if the centrifugal force were to cease. The moon 
therefore is retained in her orbit by gravity, and gravity 
only, for it would be unphilosophical to assign two causes 
to account for effects precisely similar. 

§ 364. Thus also the ratio of the forces of gravitation 
of the moon towards the sun and the earth may be esti- 
mated. For, 365J days being the periodic time of the earth 
and moon about the sun, and 27.3 days the periodic time 
of the moon about the earth ; also 60 being the distance 
of the moon from the earth in terms of the earth's radius, 
and 23,920 her mean distance from the sun in the same 

, 23920 60 -it p 09 i 

measure, we have gg-^p : g^-p : : F : f : : 2f : 1 near- 
ly ; that is, the moon's gravitation towards the sun is to 
her gravitation towards the earth as 2f to 1 nearly. 

Again, from the same principles, the centrifugal force of 
a body at the equator, arising from the rotation of the 
earth, is derived. For these propositions apply to 
centrifugal forces as well as centripetal ones, the terms, 
being correlatives (where these two alone keep the 
body in its orbit). And we have just found that the 
time of revolution is 5,075s. when the centrifugal force be- 
comes equal to the gravity ; also it appears that the forces 
in circles having the same radii are reciprocally as the 
squares of the periodic times ; hence, therefore, since the 
earth's rotation is performed in 23h. 56m., or 86,160s., 
we have 86,160 2 : 5,075 2 : : the force of gravity : the 
centrifugal force of a body at the equator arising from the 
earth's rotation : : 1 : aw nearly. 

§ 365. Since the time of revolution of a body under 
the equator, and in any parallel of latitude, is equal ; the 
centrifugal forces are as the distances from the axis of mo- 
tion, or, as the radius to the cosine of the latitude. But 
in any latitude the centrifugal force is not (as under the 



ELEMENTS OF ASTRONOMY. 265 

equator) opposite to the whole gravity, but only a part of 
it ; which also is to the whole as the cosine of the latitude 
to radius. 

Therefore combining these two ratios, it follows, that the 
diminution of gravity at the surface of the earth, arising 
from the centrifugal force, varies as the square of the co- 
sine of the latitude. 

The law just stated for the diminution of gravity is on 
the supposition of the earth's sphericity ; but as the polar 
axis of the earth is rather shorter than the equatorial, the 
former being to the latter nearly as 300 to 301, or what 
is technically denominated the compression being about 
-§-£o> the preceding theory is not exact. 

§ 366. The pendulum serves as an excellent measure 
of the force of gravity, for by it may be ascertained the 
distance through which a body unsupported would fall in a 
second of time. The following proportion will always give 
that space. 

As 3.1416 : 1 : : 1 second : the time of falling through 
a space equal to half the length of a pendulum beating 
seconds. 

But the spaces described by falling bodies are in the 
proportion of the squares of the times ; therefore : 

As the square of the time last found is to one second 
squared, so is half the length of the pendulum beating 
seconds to the space through which a body would fall in a 
second ; but this is the measure of the force exerted by the 
attraction of gravity. 

For example, suppose it is found by observation that the 
length of a pendulum beating seconds, in the latitude of 
London, is 39.126 inches, or 3.2605 feet; required the 
space through which a body would fall in a second in that 
latitude. 

As 3.1416 : 1 : : 1 sec. : 3183 sec, the time a body 
would take to fall through 1.63025 feet, half the length of 
the pendulum beating seconds. 

Again: As 3183*, or 10131489 sec. : l 2 sec. : : 
1.63025 feet, one half the length of such a pendulum, in 
latitude 51j° : 16.09 feet, the space through which a body 
would fall in a second, in that latitude. 
23 



266 ELEMENTS OF ASTRONOMY. 

§ 367. Suppose the shorter axis of an ellipse to di- 
minish continually, the longer axis remaining the same ; 
the ellipse will be transformed into a straight line, equal 
in length to the major axis. In all the successive ellipses 
produced by this gradual diminution of the minor axis, the 
periodic time remains unchanged, if the force acting at the 
centre remains unchanged. The ellipse may be considered 
as undistinguishable from the major axis, and the revolu- 
tion in such an ellipse as undistinguishable from the ascent 
of a body along the axis, to its subsequent descent in an 
equal time. Consequently a body solicited by such a cen- 
tral force will descend through the space in half the time 
of revolution in the ellipse. Let T be the time of revolu- 
tion of a planet at any distance, and t the time of a revo- 
lution at half that distance ; then, by the third law of 

Kepler, T 2 : t 2 : : 2 3 : l 3 ; hence, t =773, and Jt±* T7%% > 

but we have just seen that Jt is the time in which a body 
would fall to the sun from the distance corresponding to T : 
therefore, the time in which a planet would fall to the sun 
by the action of the centripetal force is equal to its pe- 
riodic time divided by ^/32 ; or it is equal to that time mul- 
tiplied by the reciprocal of ^32, that is, by 0, 176776. 
By this general rule, the times in which the different 
planets would reach the sun, if let fall when at their mean 
distances, may be determined. 



ELEMENTS OF ASTRONOMY. 267 

CHAPTER XVI. 

PERTURBATIONS.* 



Disturbing Forces. Problem of the three bodies. Stability of the Solar 
System. Periodical and Secular Inequalities. Perturbations in Longi- 
tude. Motion of the Line of Apsides. Variation of the Eccentricities. 
Perturbations in Latitude. Retrogradation of the Nodes. Variation of 
the Inclinations. Permanency of the Major Axes. Effect of a Resisting 
Medium. Invariable Plane of the Solar System. Inequality in the The- 
ory of Jupiter and Saturn. 

§ 368. We must now introduce some modification 
into the facts and the laws we have been asserting. Since 
all members of the solar system are exposed to one 
another's influence, and are free to move, they cannot 
retain the motions which the sun's influence alone would 
impress on them. Were the planets attracted by the sun 
only, they would describe perfect ellipses ; as each planet 
and satellite attracts every other planet and satellite, they 
move in no known or symmetrical curve, but in paths now 
approaching to, now receding from the elliptical form. 
Thus we find in the heavens no perfect ellipse, no immova- 
ble plane, no unvarying motion ; no cubes of the distances 
bear precisely the same proportion to the squares of the 
times, no radius vector sweeps over equal areas in exactly 
equal times. The areas really described, however, the de- 
partures from an elliptic path, the alterations of the planes, 
and the motions of the nodes and the apsides become the 
tests of the disturbing forces. 

An attraction which acts equally and in the same direc- 
tion on two bodies does not disturb their relative motions. 
The force which disturbs the motion of a satellite or a planet 
is the difference of the forces which act on the central and 
revolving body. Thus if the moon is between the sun and 
the earth, and if the sun's attraction, in a certain time, 

* This chapter is taken from Mrs. Somerville's " Connection of the Phy- 
sical Sciences." 



268 ELEMENTS OF ASTRONOMY. 

draws the earth 200 inches, and in the same time draws 
the moon 201 inches, then the real disturbing force is the 
force which would produce in the moon a motion of one 
inch from the earth. If the direction of the attracting 
force is different in the two cases, some complication is in- 
troduced ; but by the resolution of forces the amount of 
disturbance in any given direction may be found. 

The disturbing body may be exterior to the orbit of the 
revolving body, as Jupiter to the earth ; or within it, as 
Venus to the earth ; or it may be central and fixed, as the 
sun, while the two bodies whose relative motions are dis- 
turbed both revolve round it. 

§ 369. The simplest mode of considering perturba- 
tions is however to consider merely the amount of force 
and the direction in which it is exerted, without regard to 
the body exerting it. 

To determine the motion of each body, when disturbed 
by all the rest, is beyond the power of analysis. It is 
therefore necessary to estimate the disturbing action of one 
planet at a time, whence the celebrated problem of the 
three bodies, originally applied to the moon, the earth, and 
the sun ; namely, the masses being given of three bodies 
projected from three given points, with velocities given, 
both in quantity and direction ; and, supposing the bodies 
to gravitate to one another with forces that are directly as 
their masses, and inversely as the squares of their dis- 
tances, to find the lines described by these bodies, and 
their positions at any given instant : or in other words, to 
determine the path of a celestial body when attracted by a 
second body, and disturbed in its motions round the second 
body by a third — a problem equally applicable to planets, 
satellites, and comets. 

By this problem the motions of translation of the celes- 
tial bodies are determined. It is an extremely difficult 
one, and would be infinitely more so, if the disturbing ac- 
tion were not very small when compared with the central 
force ; that is, if the action of the planets on one another 
were not very small when compared with that of the sun. 
As the disturbing influence of each body may be found 
separately, it is assumed that the action of the whole sys- 



ELEMENTS OF ASTRONOMY. 269 

tern, in disturbing any one planet, is equal to the sum of 
all the particular disturbances it experiences, on the gen- 
eral mechanical principle, that the sum of any of the small 
oscillations is nearly equal to their simultaneous and joint 
effect. 

§ 370. On account of the reciprocal action of matter, 
the stability of the system depends on the intensity of the 
primitive momentum of the planets, and the ratio of their 
masses to that of the sun ; for the nature of the conic 
sections in which the celestial bodies move, depends upon 
the velocity with which they were first impelled in space. 
Had that velocity been such as to make the planets move 
in orbits of unstable equilibrium, their mutual attractions 
might have changed them into parabolas, or even hyperbo- 
las ; so that the earth and planets might, ages ago, have 
been sweeping far from our sun through the abyss of space. 
But as the orbits differ very little from circles, the momen- 
tum of the planets, when projected, must have been exactly 
sufficient to insure the permanency and stability of the 
system. Besides the mass of the sun is vastly greater 
than that of any planet ; and as their inequalities bear the 
same ratio to the elliptical motions, that their masses do 
to that of the sun, their mutual disturbances only increase 
or diminish the eccentricities of their orbits by very minute 
quantities ; consequently the magnitude of the sun's mass 
is the principal cause of the stability of the system. 
There is not in the physical world a more splendid exam- 
ple of the adaptation of means to the accomplishment of an 
end, than is exhibited in the nice adjustment of these 
forces, at once the cause of the variety and of the order of 
nature. 

§ 371. The planets are subject to disturbances of two 
kinds, both resulting from the constant operation of their 
reciprocal attraction : one kind, depending upon their posi- 
tions with regard to each other, begins from zero, increases 
to a maximum, decreases, and becomes zero again, when 
the planets return to the same relative positions. In con- 
sequence of these, the disturbed planet is sometimes drawn 
away from the sun, sometimes brought nearer to him : 
sometimes it is accelerated in its motion, sometimes re- 
23* 



270 ELEMENTS OF ASTRONOMY. 

tarded. At one time it is drawn above the plane of its 
orbit, at another time below it, according to the position of 
the disturbing body. All such changes, being accom- 
plished in short periods, some in a few months, others in 
years, or in hundreds of years, are denominated periodic 
inequalities. The inequalities of the other kind, though 
occasioned likewise by the disturbing energy of the planets, 
are entirely independent of their relative positions. They 
depend upon the relative positions of the orbits alone, 
whose forms and places in space are altered by very minute 
quantities, in immense periods of time, and are therefore 
called secular inequalities. 

The periodical disturbances are compensated, when the 
bodies return to the same relative positions with regard to 
one another and the sun : the secular inequalities are com- 
pensated, when the orbits return to the same positions 
relatively to one another, and to the plane of the ecliptic. 

§ 372. Planetary motion, including both these kinds 
of disturbance, may be represented by a body revolving in 
an ellipse, and making small and transient deviations, now 
on one side of its path, and now on the other, whilst the 
ellipse itself is slowly, but perpetually changing both in 
form and position. 

The periodic inequalities are merely transient deviations 
of a planet from its path, the most remarkable of which 
only lasts about 918 years ; but in consequence of the 
secular disturbances, the apsides, or extremities of the 
major axis of all the orbits, have a direct but variable mo- 
tion in space, excepting those of the orbit of Venus, which 
are retrograde, and the lines of the nodes move with a 
variable velocity in a contrary direction. Besides these, 
the inclination and eccentricity of every orbit are in a state 
of perpetual but slow change. These effects result from 
the disturbing influence of all the planets on each. But as 
it is only necessary to estimate the disturbing influence of 
one body at a time, what follows may convey some idea of 
the manner in which one planet disturbs the elliptical mo- 
tion of another. 

§ 373. Suppose two planets moving in ellipses round 
the sun ; if one of them attracted the other and the sun 



ELEMENTS OF ASTRONOMY. 271 

with equal intensity, and in parallel directions, it would 
have no effect in disturbing the elliptical motion. The in- 
equality of this attraction is the sole cause of perturbation, 
and the difference between the disturbing planet's action 
on the sun and on the disturbed planet constitutes the dis- 
turbing force, which consequently varies in intensity and 
direction with every change in the relative positions of the 
three bodies. Although both the sun and planet are under 
the influence of the disturbing force, the motion of the dis- 
turbed planet is referred to the centre of the sun as a fixed 
point, for convenience. The whole force which disturbs a 
planet, is equivalent to three partial forces. One of these 
acts on the disturbed planet, in the direction of a tangent 
to its orbit, and is called the tangential force ; it occasions 
secular inequalities in the form and position of the orbit in 
its own plane, and is the sole cause of the periodical per- 
turbations in the planet's longitude. Another acts upon 
the same body in the direction of its radius vector, that is, 
in the line joining the centres of the sun and planet, and 
is called the radial force : it produces periodical changes 
in the distance of the planet from the sun, and affects the 
form and position of the orbit in its own plane. The third, 
which may be called the perpendicular force, acts at right 
angles to the plane of the orbit, occasions the periodic ine- 
qualities in the planet's latitude, and affects the position of 
the orbit with regard to the plane of the ecliptic. 

§ 374. It has been observed, that the radius vector, 
of a planet moving in a perfectly elliptical orbit, passes 
over equal spaces or areas in equal times ; a circumstance 
which is independent of the law of the force, and would 
be the same, whether it varied inversely as the square of 
the distance, or not, provided only that it be directed to 
the centre of the sun. Hence the tangential force, not 
being directed to the centre, occasions an unequable de- 
scription of areas, or what is the same thing, it disturbs 
the motion of the planet in longitude. The tangential force 
sometimes accelerates the planet's motion, sometimes re- 
tards it, and occasionally has no effect at all. Were the 
orbits of both planets circular, a complete compensation 
would take place at each revolution of the two planets, be- 



272 ELEMENTS OP ASTRONOMY. 

cause the arcs in which the accelerations and retardations 
take place, would be symmetrical on each side of the dis- 
turbing force. For it is clear, that if the motion be accel- 
erated through a certain space, and then retarded through 
as much, the motion at the end of the time will be the 
same as if no change had taken place. But as the orbits 
of the planets are ellipses, this symmetry does not hold : 
for, as the planet moves unequably in its orbit, it is in some 
positions more directly, and for a longer time, under the 
influence of the disturbing force than in others. And al- 
though multitudes of variations do compensate each other 
in short periods, there are others, depending on peculiar 
relations among the periodic times of the planets, which 
do not compensate each other till after one, or even till 
after many revolutions of both bodies. A periodical ine- 
quality of this kind in the motions of Jupiter and Saturn, 
has a period of no less than 918 years. 

§ 375. The radial force, or that part of the disturb- 
ing force which acts in the direction of the line joining the 
centres of the sun and disturbed planet, has no effect on 
the areas, but is the cause of periodical changes of small 
extent in the distance of the planet from the sun. It has 
already been shown, that the force producing perfectly 
elliptical motion varies inversely as the square of the dis- 
tance, and that a force following any other law, would 
cause the body to move in a curve of a very different kind. 
Now, the radial disturbing force varies directly as the 
distance ; and as it sometimes combines with and increases 
the intensity of the sun's attraction for the disturbed body, 
and at other times opposes and consequently diminishes it, 
in both cases it causes the sun's attraction to deviate from 
the exact law of gravity, and the whole action of this com- 
pound central force on the disturbed body, is either greater 
or less than is requisite for perfectly elliptical motion. 
When greater, the curvature of the disturbed planet's path 
on leaving its perihelion, or point nearest the sun, is 
greater than it would be in the ellipse, which brings the 
planet to its aphelion, or point farthest from the sun, be- 
fore it has passed through 180°, as it would do if undis- 
turbed. So that in this case the apsides, or extremities of 



ELEMENTS OF ASTRONOMY. 273 

the major axis, advance in space. When the central force 
is less than the law of gravity requires, the curvature of 
the planet's path is less than the curvature of the ellipse. 
So that the planet, on leaving its perihelion, would pass 
through more than 180° before arriving at its aphelion, 
which causes the apsides to recede in space. Cases both 
of advance and recess occur during a revolution of the two 
planets ; but those in which the apsides advance, prepon- 
derate. 

§ 376. This, however, is not the full amount of the 
motion of the apsides ; part arises, also, from the tangen- 
tial force, which alternately accelerates and retards the 
velocity of the disturbed planet. An increase in the 
planet's tangential velocity diminishes the curvature of its 
orbit, and is equivalent to a decrease of central force. On 
the contrary, a decrease of the tangential velocity, which 
increases the curvature of the orbit, is equivalent to an 
increase of central force. These fluctuations, owing to the 
tangential force, occasion an alternate recess and advance 
of the apsides, after, the manner already explained. An 
uncompensated portion of the direct motion arising from 
this cause, conspires with that already impressed by the 
radial force, and in some cases, even nearly doubles the 
direct motion of these points. The motion of the apsides 
may be represented by supposing a planet to move in an 
ellipse, while the ellipse itself is slowly revolving about the 
sun in the same plane. This motion of the major axis, 
which is direct in all the orbits except that of the planet 
Venus, is irregular, and so slow that it requires more than 
109,830 years, for the major axis of the earth's orbit to 
accomplish a sidereal revolution, that is, to return to the 
same stars ; and 20,984 years to complete its tropical revo- 
lution, or to return to the same equinox. The difference 
between these two periods arises from a retrograde motion 
in the equinoctial point, which meets the advancing axis, 
before it has completed its revolution with regard to the 
stars. The major axis of Jupiter's orbit requires no less 
than 200,610 years to perform its sidereal revolution, and 
22,748 years to accomplish its tropical revolution, from the 
disturbing action of Saturn alone. 



274 ELEMENTS OF ASTRONOMY. 

§ 377. A variation in the eccentricity of the disturb- 
ed planet's orbit, is an immediate consequence of the devi- 
ation from elliptical curvature, caused by the action of the 
disturbing force. When the path of the body, in proceed- 
ing from its perihelion to its aphelion, is more curved than 
it ought to be, from the effect of the disturbing forces, it 
falls within the elliptical orbit, the eccentricity is dimin- 
ished, and the orbit becomes nearly circular ; when that 
curvature is less than it ought to be, the path of the planet 
falls without the elliptical orbit, and the eccentricity is in- 
creased ; during these changes, the length of the major 
axis is not altered, the orbit only bulges out or becomes 
more flat. Thus the variation in the eccentricity arises 
from the same cause that occasions the motion of the 
apsides. There is an inseparable connection between these 
two elements : they vary simultaneously, and have the same 
period ; so that whilst the major axis revolves in an im- 
mense period of time, the eccentricity increases and de- 
creases by very small quantities, and at length returns to 
its original magnitude at each revolution of the apsides. 
The terrestrial eccentricity is decreasing at the rate of 
about forty miles annually; and if it were to decrease 
equably, it would be 39,861 years before the earth's orbit 
became a circle. The mutual action of Jupiter and Saturn 
occasions variations in the eccentricities of both orbits, the 
greatest eccentricities of Jupiter's orbit corresponding to 
the least of Saturn's. The period in which these vicissi- 
tudes are accomplished is 70,414 years, estimating the ac- 
tion of these two planets alone ; but if the action of all 
the planets were estimated, the cycle would extend to mil- 
lions of years. 

§ 378. That part of the disturbing force is now to be 
considered which acts perpendicularly to the plane of the 
orbit, causing periodic perturbations in latitude, secular 
variations in the inclination of the orbit, and a retrograde 
movement to its nodes on the true plane of the ecliptic. 
This force tends to pull the disturbed body above, or push 
it below the plane of its orbit, according to the relative po- 
sitions of the two planets with regard to the sun, considered 
to be fixed. By this action it sometimes makes the plane 



ELEMENTS OP ASTRONOMY. 275 

of the orbit of the disturbed body tend to coincide with the 
plane of the ecliptic, and sometimes increases its inclina- 
tion to that plane. In consequence of which, its nodes al- 
ternately recede or advance on the ecliptic. When the 
disturbing planet is in the line of the disturbed planet's 
nodes, it neither affects these points, the latitude, nor the 
inclination, because both planets are then in the same plane. 
When it is at right angles to the line of the nodes, and the 
orbit symmetrical on each side of the disturbing force, the 
average motion of these points, after a revolution of the 
disturbed body, is retrograde, and comparatively rapid ; 
but when the disturbing planet is so situated that the orbit 
of the disturbed planet is not symmetrical on each side of 
the disturbing force, which is most frequently the case, 
every possible variety of action takes place. Consequently, 
the nodes are perpetually advancing or receding with une- 
qual velocity ; but as a compensation is not effected, their 
motion is, on the whole, retrograde. 

§ 379. With regard to the variations in the inclina- 
tion, it is clear, that, when the orbit is symmetrical on each 
side of the disturbing force, all its variations are compen- 
sated after a revolution of the disturbed body, and are 
merely periodical perturbations in the planet's latitude ; 
and no secular change is induced in the inclination of the 
orbit. When, on the contrary, that orbit is not symmetri- 
cal on each side of the disturbing force, although many of 
the variations in latitude are transient or periodical, still, 
after a complete revolution of the disturbed body, a portion 
remains uncompensated, which forms a secular change in 
the inclination of the orbit to the plane of the ecliptic. It 
is true, part of this secular change in the inclination is 
compensated by the revolution of the disturbing body, 
whose motion has not hitherto been taken into the account, 
so that perturbation compensates perturbation ; but ^ still, a 
comparatively permanent change is effected in the inclina- 
tion, which is not compensated till the nodes have accom- 
plished a complete revolution. 

§ 380. The changes in the inclination are extremely 
minute, compared with the motion of the nodes, and there 
is the same kind of inseparable connection between their 



276 ELEMENTS OF ASTRONOMY. 

secular changes that there is between the variation of the 
eccentricity and the motion of the major axis. The nodes 
and inclinations vary simultaneously, their periods are the 
same, and very great. The nodes of Jupiter's orbit, from 
the action of Saturn alone, require 36,261 years to accom- 
plish even a tropical revolution. In what precedes, the in- 
fluence of only one disturbing body has been considered ; 
but when the action and reaction of the whole system are 
taken into account, every planet is acted upon, and does 
itself act, in this manner, on all the others ; and the joint 
effect keeps the inclinations and eccentricities in a state of 
perpetual variation. It makes the major axes of all the 
orbits continually revolve, and causes, on an average, a 
retrograde motion of the nodes of each orbit upon every 
other. The ecliptic itself is in motion from the mutual ac- 
tion of the earth and planets, so that the whole is a com- 
pound phenomenon of great complexity, extending through 
unknown ages. At the present time, the inclinations of all 
the orbits are decreasing, but so slowly, that the inclination 
of Jupiter's orbit is only about six minutes less than it was 
in Ptolemy's time. 

§ 381. But, in the midst of all these vicissitudes, the 
length of the major axes and the mean motions of the 
planets remain permanently independent of secular changes. 
They are so connected by Kepler's law, of the squares of 
the periodic times being proportional to the cubes of the 
mean distances of the planets from the sun, that one cannot 
vary without affecting the other. And it is proved that any 
variations which do take place are transient, and depend 
only on the relative positions of the bodies. 

It is true that, according to theory, the radial disturbing 
force should permanently alter the dimensions of all the 
orbits, and the periodic times of all the planets, to a cer- 
tain degree. For example, the masses of all the planets 
revolving within the orbit of any one, such as Mars, by 
adding to the interior mass, increase the attracting force 
of the sun, which, therefore, must contract the dimensions 
of the orbit of that planet, and diminish its periodic time ; 
whilst the planets exterior to the orbit of Mars must have 
the contrary effect. But the mass of the whole of the 



ELEMENTS OE ASTRONOMY. 277 

planets and satellites taken together is so small, when com- 
pared with that of the sun, that these effects are quite in- 
sensible, and could only have been discovered by theory. 
And, as it is certain that the length of the major axis and 
mean motions are not permanently changed by any other 
power whatever, it may be concluded that they are inva- 
riable. 

§ 382. "With the exception of these two elements, it 
appears that all the bodies are in motion, and every orbit 
in a state of perpetual change. Minute as these changes 
are, they might be supposed to accumulate, in the course 
of ages, sufficiently to derange the whole order of nature, 
to alter the relative positions of the planets, to put an end 
to the vicissitudes of the seasons, and to bring about colli- 
sions, which would involve our whole system, now so har- 
monious, in chaotic confusion. It is natural to inquire 
what proof exists that nature will be preserved from such a 
catastrophe ? Nothing can be known from observation, 
since the existence of the human race has occupied com- 
paratively but a point in duration, while these vicissitudes 
embrace myriads of ages. The proof is simple and conclu- 
sive. All the variations of the solar system, secular as 
well as periodic, are expressed analytically by the sines 
and cosines of circular arcs, which increase with the time ; 
and, as a sine or cosine can never exceed the radius, but 
must oscillate between zero and unity, however much the 
time may increase, it follows that, when the variations 
have accumulated to a maximum, by slow changes, in how- 
ever long a time, they decrease, by the same slow degrees, 
till they arrive at their smallest value, again to begin a 
new course ; thus forever oscillating about a mean value. 
This circumstance, however, would be insufficient were it 
not for the small eccentricities of the planetary orbits, 
their minute inclinations to the plane of the ecliptic, and 
the revolutions of all the bodies as well planets as satel- 
lites in the same direction. These secure the perpetual 
stability of the solar system. 

§ 383. The equilibrium, however, would be deranged, 
if the planets moved in a resisting medium sufficiently 
dense to diminish their tangential velocity, for then both 
24 



278 ELEMENTS OF ASTRONOMY. 

the eccentricities and the major axes of the orbits would 
vary with the time, so that the stability of the system 
would be ultimately destroyed. The existence of an ethe- 
rial fluid is now proved ; and, although it is so extremely 
rare that hitherto its effects on the motions of the planets 
have been altogether insensible, there can be no doubt, 
that, in the immensity of time, it will modify the forms of 
the planetary orbits, and may at last even cause the de- 
struction of our system, which in itself contains no princi- 
ple of decay, unless a rotary motion from west to east has 
been given to this fluid by the bodies of the solar system, 
which have all been revolving about the sun in that direc- 
tion for unknown ages. This rotation, which seems to be 
highly probable, may even have been coeval with its crea- 
tion. 

§ 384. The form and position of the planetary orbits, 
and the motion of the bodies in the same direction, to- 
gether with the periodicity of the terms in which the ine- 
qualities are expressed, assure us that the variations of the 
system are confined within very narrow limits, and that al- 
though we do not know the extent of the limits, nor the 
period of that grand cycle, which probably embraces mil- 
lions of years, yet they never will exceed what is requisite 
for the stability and harmony of the whole, for the preser- 
vation of which every circumstance is so beautifully and 
wonderfully adapted. 

The plane of the ecliptic itself, though assumed to be 
fixed at a given epoch for the convenience of astronomical 
computation, is subject to a minute secular variation of 
45" 7 ; occasioned by the reciprocal action of the planets. 
But as this is also periodical, and cannot exceed 2° 42 r , 
the terrestrial equator, which is inclined to it at an angle of 
23° 27' 37" 89, will never coincide with "the plane of the 
ecliptic ; so there never can be perpetual spring. The ro- 
tation of the earth is uniform ; therefore day and night, 
summer and winter, will continue their vicissitudes, while 
the system endures, or is undisturbed by foreign causes. 

§ 385. Notwithstanding the permanency of our sys- 
tem, the secular variations in the planetary orbits would 
have been extremely embarrassing to astronomers when it 



ELEMENTS OF ASTRONOMY. 279 

became necessary to compare observations separated by 
long periods. The difficulty was in part obviated, and the 
principle for accomplishing it established by La Place, and 
has since been extended by M. Poinsot. It appears that 
there exists an invariable plane, passing through the centre 
of gravity of the system, about which the whole oscillates 
within very narrow limits, and that this plane will always 
remain parallel to itself, whatever changes time may in- 
duce in the orbits of the planets, in the plane of the eclip- 
tic, or even in the law of gravitation ; provided only that 
our system remains unconnected with any other. La Place 
found that the plane in question is inclined to the ecliptic 
at an angle of nearly 1° 34' 15", and that, in passing 
through the sun, and about midway between the orbits of 
Jupiter and Saturn, it may be regarded as the equator of 
the solar system, dividing it into two parts, which balance 
one another in all their motions. This plane of greatest 
inertia, by no means peculiar to the solar system, but ex- 
isting in every system of bodies submitted to their mutual 
attractions only, always maintains a fixed position, whence 
the oscillations of the system may be estimated through 
unlimited time. 

§ 386. Future astronomers will know, from its immu- 
tability or variation, whether the sun and his attendants 
are connected or not with the other systems of the universe. 
Should there be no link between them, it may be inferred, 
from the rotation of the sun, that the centre of gravity of 
the system situate within his mass describes a straight line 
in this invariable plane or great equator of the solar sys- 
tem, which, unaffected by the changes of time, will main- 
tain its stability through endless ages. But, if the fixed 
stars, comets, or any unknown and unseen bodies, affect 
our sun and planets, the nodes of this plane will slowly re- 
cede on the plane of that immense orbit which the sun may 
describe about some most distant centre, in a period which 
it transcends the power of man to determine. There is 
every reason to believe that this is the case ; for it is more 
than probable that, remote as the fixed stars are, they in 
some degree influence our system, and that even the inva- 
riability of this plane is relative, only appearing to be fixed 



280 ELEMENTS OF ASTRONOMY. 

to creatures incapable of estimating its minute and slow 
changes during the small extent of time and space granted 
to the human race. If we raise our views to the whole 
extent of the universe, and consider the stars together with 
the sun, to be wandering bodies, revolving about the com- 
mon centre of creation, we may then recognise in the equa- 
torial plane passing through the centre of gravity of the 
universe the only instance of absolute and eternal repose. 

§ 387. All the periodic and secular inequalities de- 
duced from the law of gravitation, are so perfectly con- 
firmed by observation, that analysis has become one of the 
most certain means of discovering the planetary irregular- 
ities, either when they are too small, or too long in their 
periods to be detected by other methods. Jupiter and 
Saturn, however, exhibit inequalities, which for a long time 
seemed discordant with that law. All observations, from 
those of the Chinese and Arabs down to the present day, 
prove that for ages the mean motions of Jupiter and Saturn 
have been affected by a great inequality of a very long pe- 
riod, forming an apparent anomaly in the theory of planets. 
It was long known, by observation, that five times the mean 
motion of Saturn is nearly equal to twice that of Jupiter ; 
a relation which the sagacity of La Place perceived to be 
the cause of a periodic irregularity in the mean motion of 
each of these planets, which completes its period in 918 
years, the one being retarded while the other is accelerated ; 
but both the magnitude and period of these quantities vary, 
in consequence of the secular variations in the elements of 
the orbits. Suppose the two planets to be on the same 
side of the sun, and all three in the same straight line, 
they are then said to be in conjunction. Now if they be- 
gin to move at the same time, one making exactly five re- 
volutions in its orbit, while the other only accomplishes two, 
it is clear that Saturn, the slow moving body, will only 
have got through a part of its orbit during the time that 
Jupiter has made one whole revolution and part of another, 
before they be again in conjunction. 

§ 388. It is found that during this time their mutual 
action is such as to produce a great many perturbations 
which compensate each other, but there still remains a por- 



ELEMENTS OP ASTRONOMY. 281 

tion outstanding, owing to the length of time during which 
the forces act in the same manner ; and if the conjunction 
always happened in the same point of the orbit, this un- 
compensated inequality in the mean motion would go on in- 
creasing till the periodic times and forms of the orbits were 
completely and permanently changed ; a case that would 
actually take place if Jupiter accomplished exactly five re- 
volutions in the time Saturn performed two. These revo- 
lutions are, however, not exactly commensurable ; the points 
in which the conjunctions take place are in advance each 
time as much as 8°. 37 ; so that the conjunctions do not 
happen exactly in the same points of the orbits till after a 
period of 850 years ; and in consequence of this small ad- 
vance, the planets are brought into such relative positions 
that the inequality which seemed to threaten the stability 
of the system is completely compensated, and the bodies 
having returned to the same relative positions with regard 
to one another and the sun, begin a new course. The 
secular variations in the elements of the orbit increase the 
period of the inequality to 918 years. As any perturba- 
tion which affects the mean motion affects also the major 
axis, the disturbing forces tend to diminish the major axis 
of Jupiter's orbit, and increase that of Saturn's during one 
half of the period, and the contrary during the other half. 
This inequality is strictly periodical, since it depends on 
the configuration of the two planets ; and theory is con- 
firmed by observation, which shows that, in the course of 
twenty centuries, Jupiter's mean motion has been accele- 
rated by about 3° 23', and Saturn's retarded by 5° 13'. 

§ 389. Several instances of perturbations of this kind 
occur in the solar system. One, in the mean motions of 
the Earth and Venus only amounting to a few seconds, has 
been recently worked out with immense labor by Professor 
Airy. It accomplishes its changes in 240 years, and arises 
from the circumstance of thirteen times the periodic time 
of Venus being nearly equal to eight times that of the 
earth. Small as it is, it is sensible in the motions of the 
earth. 

It might be imagined that the reciprocal action of such 
planets as have satellites would be different from the influ- 
24* 



282 ELEMENTS OF ASTRONOMY. 

ence of those that have none. But the distances of the 
satellites from their primaries are incomparably less than 
the distances of the planets from the sun, and from one 
another. So that the system of a planet and its satellites, 
moves nearly as if all these bodies were united in their 
common centre of gravity. The action of the sun, how- 
ever, in some degree disturbs the motion of the satellites 
about their primary. 



CHAPTER XVII 



NUTATION, AND ABERRATION. 

Action of the Planets on the Plane of the Ecliptic. Action of the Sun and 
Moon on the Earth's Equator. The Precession of the Equinoxes. 
Motion of the Earth's Axis. Nutation. Aberration. Its Effect on the 
Apparent Places of the Stars. Methods of Computing it. Aberration of 
the Fixed Stars and of the Planets. 

§ 390. The planets acting on the sun and on the earth, 
as a whole, occasion a slow variation in the plane of the 
ecliptic, which affects its inclination to the plane of the 
equator, and but for other causes would make the equator 
cross the ecliptic every year 0/ r 31 in advance of the last 
equinox. The disturbing influence of Venus and Jupiter 
in particular on the earth diminishes the obliquity of the 
ecliptic annually by 0"457. This variation, in the course 
of ages, may amount to 10° or 11° ; but the obliquity of 
the ecliptic to the equator can never vary more than 2° or 
3°, since the equator will follow in some degree the motion 
of the ecliptic. 

But while the action of the planets would cause the 
equator to cross the ecliptic later every year, more powerful 
influences are drawing them to intersect earlier. The earth, 
it will be remembered, is not a perfect sphere. The sun and 
moon being always, one in, and the other near, the plane 
of the ecliptic, act obliquely and unequally on different 
parts of the spheroid, and urge the plane of the equator 



ELEMENTS OP ASTRONOMY. 283 

backward from east to west. There are but two positions 
(in the equinoxes) when the sun does not urge the earth's 
equator to change its position. At all other times, by its 
action on the protuberant matter at the equator, it tends to 
draw the equatorial parts under itself, and thus causes a 
balancing of the equator. 

§ 391. The direct tendency of this attraction is to 
make the planes of the equator and ecliptic coincide. It is 
difficult to imagine or compute the effects of this tendency, 
for the rapid rotation of the earth continually presents dif- 
ferent parts to the sun, and thus changes the direction, 
and consequently the force of the sun's attraction on the 
equatorial parts. And the equatorial parts cannot obey 
the attraction without drawing along the whole mass of the 
earth. Therefore, the motion of the whole earth, which 
can be caused by the excess of attraction on the equatorial 
parts at one moment, is extremely small. 

The inclination of the planes is not affected by this cause ; 
but in the course of the year sufficient attraction is exer- 
cised by the sun to cause the equator to cut the ecliptic in 
the vernal equinox, at a point 15" sooner than it would if 
the earth were a perfect sphere. This recession of the 
equator causes the precession of the equinoxes ; that is, 
causes the equinox to occur sooner than it would if the 
earth were a perfect sphere. If the flattening of the earth 
were greater than it is, precession would also increase. 

The orbit of the moon being inclined 5°. 8 to the ecliptic, 
is sometimes inclined 29° to the earth's equator. The 
moon's action on the redundant matter at the equator has 
a similar effect to that of the sun. 

§ 392. The actual yearly recession caused by the 
sun and moon would be rather more than 50" 22, but it is 
partly balanced by the action of the planets mentioned 
before. Thus the sun, moon and planets, by moving the 
plane of the equator, cause the equinoctial points to retro- 
grade on the ecliptic, without however in the whole chang- 
ing the angle made by the equator and ecliptic. And 
the planets move the plane of the ecliptic, and give the 
equinoctial points a direct motion, though much less than 
the former. 



284 ELEMENTS OF ASTRONOMY. 

The direct action of the planets alone on the equatorial 
parts of the earth causes a slight retrogradation of 50" a 
century. 

I have mentioned that even while the planets act on the 
earth independently of its figure, they have an indirect 
influence connected with this figure. By displacing the 
plane of the ecliptic, they bring the sun and moon into dif- 
ferent positions with respect to the earth, and thus modify 
their action on the equatorial parts. These inequalities 
render the retrogradation unequal in different centuries. 

The period in which the equinoctial points would accom- 
plish an entire revolution in the ecliptic, cannot therefore 
be precisely fixed. It does not vary much from 25,868 
years. 

§ 393. If the equator retreats on the ecliptic, the 
pole of the equator must move also, and describe round 
the pole of the ecliptic from east to west, a small circle 
with a radius of 23° 28', in 25,868 years. 

If the attraction of the sun and moon were always exer- 
cised equally, this would be.the path described by the pole 
of the equator. But the inclination of the moon's orbit to 
the earth's equator, and the position of her nodes are con- 
tinually changing. She is also rapidly changing place in 
her orbit, and her orbit is changing its line of apsides, and 
consequently its point of nearest approach to the earth. 
The position of the nodes, however, has more influence 
than the place of the moon in her orbit. For, by this po- 
sition, we ascertain in which direction the moon's attraction 
on the equatorial parts will preponderate in the course of 
a lunar month. To this balancing motion of the equator, 
and the wavering motion of the pole, caused by the inequal- 
ities of the moon's action, the name of nutation is given. 
Were the pole of the earth influenced by the disturbing 
action of the moon only, it would describe about the 
pole of the moon's orbit a small ellipse, the axes of which 
are 18" 5, and 13" 6, the longer being directed toward 
the pole of the ecliptic. If this were described alone its 
period would be 19 years, the time occupied by the nodes 
of the lunar orbit in accomplishing a revolution. 

§ 394. To give an idea of the extreme minuteness of 



ELEMENTS OF ASTRONOMY. 285 

the nutation of the earth's axis caused by the moon, let us 
suppose an iron rod 100 feet long, fixed at one end and 
movable at the other, to represent one half of the earth's 
axis. If the movable end were pulled the twentieth part 
of an inch to one side, the deviation would be proportional- 
ly as great as that which the lunar nutation produces in 
the terrestrial axis. There is also a slight inequality which 
depends on the position of the sun only, which goes through 
all its values in the course of half a revolution of the sun. 
On account of it the pole would describe an ellipse, whose 
semi-major axis would be 0" 435, its semi-minor axis 
0" 399. 

The curve really traced in the heavens by the prolonga- 
tion of the earth's axis is compounded of these three mo- 
tions. While in virtue of the moon's action it would de- 
scribe a little ellipse, it is carried over so much of its cir- 
cle round the pole of the ecliptic as corresponds to 19 
years ; that is to say, over an angle of nineteen times 50" 
round the centre. The path which it will describe in vir- 
tue of these three motions will be neither an ellipse nor an 
exact circle, but a slightly undulating ring. In the follow- 
ing figure the ellipse caused by the sun's action is not 
represented. 

§ 395. Let C, (Plate II. Fig. 4), be the centre of the 
earth, C P half its axis, P the north pole, and A S B half of 
the equator. Let m n be part of the plane of the ecliptic, 
and C Q a line perpendicular to it, pointing therefore to 
the pole of the ecliptic in the heavens. If then P be car- 
ried uniformly round a circle perpendicular to C Q, so that 
C P shall describe a conical surface, the equinoxes B and A 
will be carried round in a direction contrary to that of the 
diurnal motion, and with them the equator B S A, the angle 
which the equator makes with the ecliptic remaining unal- 
tered. This motion of B and A is the precession. But 
suppose that instead of P's being placed on the circle, it is 
placed on the circumference of a small oval which has its 
centre on the circle. While the centre of the oval moves 
forward on the circle with the motion of precession, let the 
pole P move round the oval with a motion much slower 



286 ELEMENTS OF ASTRONOMY. 

than that of precession. It will then trace out in space an 
undulating curve, and there will be an alternate retarda- 
tion and acceleration of the motion of the equinoxes along 
the plane of the ecliptic, together with a vibration of the 
plane of the equator to and from the ecliptic, which are the 
motions constituting nutation. 

§ 398. We must now make a slight change in our 
conception of the motion of the earth, and imagine its cen- 
tre moving on in the ecliptic, while at the same time all 
the parts but the axis of the equator rotate. This axis 
is not wholly without motion. Its middle point, to be sure, 
is always stationary* but the ends of the axis describe two 
circles blended, one owing to precession, one owing to 
nutation ; the cause of these motions we have just seen ; 
let us now view them separately, without considering the 
especial share which the sun, moon, and planets have in 
each. 

If our eyes were keen enough to discern exceedingly 
slow motion, or if we could see at one glance the motion of 
years, we should see the equinoxes moving back on the 
ecliptic, with a variable motion. At the same time the 
equator would appear to swing backward and forward 
to and from the ecliptic, turning upon the equinoxes 
as pivots. Of these motions the average motion of the 
equator on the ecliptic is the precession ; the alternate ac- 
celeration and retardation are one part of the nutation ; 
and the alternate increase and diminution of the angle con- 
tained between the two, is the other part. 

§ 397. As the fixed stars do not alter their positions 
among themselves in consequence of the retrocession of the 
ecliptic, by observing when the equator passes through 
them, it is easy to ascertain the amount of its motion. 
This being known, we know the position of the vernal equi- 
nox at that time, a point which it is essential to know, 
since we reckon from it longitudes and right ascensions, 
and also begin there our equinoctial year. When we speak 
of the right ascension or declination of any celestial object, 
we must therefore mention what epoch we intend. 

Precession affects the longitudes of all the stars, their 
declinations and right ascensions, but not their latitudes. 



ELEMENTS OP ASTRONOMY. 287 

The right ascensions and the declinations of different 
stars are variously affected by the motion of the pole of 
the equator. Some stars and constellations are brought 
by precession near to the pole, and others appear to recede 
from it. Its effects on the places of the stars are so strik- 
ing, that it was discovered more than a century before the 
Christian era, though its cause remained unknown. Nu- 
tation, which makes but inconsiderable changes among the 
stars, has been known but little more than a hundred 
years. Nutation causes an apparent approach and recess 
of all the stars in the heavens to the pole in the period of 
nineteen years. The equinoctial points have also a small 
alternate balancing motion in the 'same period, by which 
the longitudes and right ascensions of the stars are alter- 
nately increased and diminished. 

§ 398. In the year 158 before Christ, Hipparchus dis- 
covered precession from the comparison of his own observa- 
tions with those made 155 years before. He had formed 
a catalogue in which he laid down the latitude and longi- 
tude of every star ; and he supposed that this work once 
performed would give the true places of the stars forever. 
In his own lifetime, however, he found that all the fixed 
stars were sweeping with a very slow motion towards the 
east, and that while their latitudes remained the same, their 
longitudes would increase the 360th part of the whole cir- 
cumference in 72 years. He had no hesitation between 
attributing a real motion to the stars or one in the opposite 
direction to the earth. 

§ 399. It is shown, both by observation and theory, 
that the orbits of all the planets, as well as that of the 
earth, have this motion by which their nodes move on the 
ecliptic, and on the imaginary plane of inertia before men- 
tioned. They do not, however, all retreat on the plane of 
inertia. Those whose orbits are most inclined to this 
plane are drawn down by those whose orbits are less in- 
clined , and .their nodes advance ; but those whose orbits 
are more nearly in this plane, are drawn from it by those 
which are farther removed, and their nodes consequently 
retreat. Even those orbits which actually retreat on the 



288 ELEMENTS OF ASTRONOMY. 

plane of inertia, may appear to advance on that part of 
any orbit which lies above this plane. 

In all the planets, except Venus, there is a very little 
more than a complete revolution between two aphelia ; in 
Venus, there is a little less. The apparent annual motion 
of the aphelion of each planet's orbit is the motion arising 
from precession plus that arising from the motion of the 
apsides. In Venus, the motion arises from precession minus 
the motion of the apsides. The apparent motion of Venus's 
aphelion is like that of the other planets ; for though the 
aphelion moves backward, the equinox does the same at a 
greater rate. The real motion of its apsides is regressive, 
because its orbit is very nearly circular, and the amount of 
the earth's influence is to make it regress. 

§ 400. Precession and nutation aiford us some light as 
to the condition and density of the interior of the earth. 
The mean density of the earth being little more than 5 j- 
times the density of water, and the rocks on the surface 
averaging only about 2|- times the weight of water, it fol- 
lows that the interior must be as much above the average 
as the surface is below it. 

The ring of matter round the equator being composed 
partly of rock and partly of water, cannot average above 
2 J times the weight of water. Its effect in disturbing the 
earth's axis, is therefore slighter than if it were of the same 
density with the interior. The disturbance found by cal- 
culation is less than if the earth were a homogenious sphe- 
roid, we are therefore justified in believing it not to be one. 
We find, also, that the comparative densities of the interior 
and exterior, found by the disturbance of a pendulum near 
a mountain, agree with those inferred from nutation. 

The observed amount of precession seems to require for 
the crust of the globe a greater thickness than many geolo- 
gists have allowed. It requires a thickness of at least one 
fourth or one fifth of the earth's radius ; that is, from eight 
hundred to a thousand miles. It does not forbid the sup- 
position that the earth is solid to the centre. 

§401. There is another phenomenon besides the change 
in longitude of the stars, which early made the precession 



ELEMENTS OE ASTRONOMY. 289 

of the equinoxes suspected. The sun was found to cross 
the equinox before he returned to the same stars. In other 
words, the equinoctial was found to be shorter than the 
sidereal year. If there were no precession, the equinoctial 
and sidereal year would agree. As it exists, the equinoc- 
tial year of 365d. 5h. 48m. 49s. must be increased by the 
time the sun takes to move through an arc of 50"22, in 
order to have the length of the sidereal year. The time 
required is 20' 19" ; so that the sidereal year contains 
365d. 6h. 9m. 9s., mean solar days. 

Owing to the variations in the action of the sun, and the 
consequent inequalities in the precession of the equinoxes, 
the equinoctial year is four or five seconds shorter than it 
was in the time of Hipparchus. The annual retrogradation 
of the equinoctial points being greater by OZ'455 than it 
was in his time, the sun has each year a space of 0."455 
less in the ecliptic to pass through, in order to reach the 
plane of the equator. The utmost change in the length of 
the year from this cause is 43 seconds. 

§ 402. Besides the effect of precession and nutation, 
another cause influences the apparent positions of the stars. 
This is the aberration of light. 

We judge of the position of bodies by the direction of 
rays which enter our eyes, and which appear to proceed 
from them. We know not how often these rays have been 
reflected or refracted, whether they have been left behind 
by some body rushing on in its course, or whether some 
motion of our own has caused them to meet our eye in 
a false direction. A ray from a lantern suspended above 
our heads may reach us at the same moment and from the 
same direction with a ray from a star, and the two objects 
will be referred to the same place. The lantern partakes 
our motion, and is seen in its true place. But the rays 
from the star are, by our motion, referred to a false direc- 
tion. 

§ 403. Even when we are conscious of our motion we 
are apt to transfer it to surrounding objects. A man walk- 
ing fast in a vertical rain, attributes to the rain his own 
motion. If he goes toward the north, the rain appears to 
him to slant toward the south. If he moves as fast as the 
25 



290 ELEMENTS OF ASTRONOMY. 

rain falls, he gives to the rain as much motion southward 
as it has vertically, and it strikes him as if it fell at an 
angle of 45°. A train of cars moving rapidly against a 
driving rain, gives its own motion to the drops, and they 
appear to the passengers more slanting than they really 
are. If the cars move with the rain, they meet the drops 
sooner than if at rest, and their apparent obliquity is less. 
Whatever may be the direction in which drops would fall 
upon cars at rest, they will meet the cars in motion in a 
different one. By the same reasoning, we see that what- 
ever may be the direction in which rays would pass from a 
star to an observer at rest, they must meet an observer in 
motion in a different one. 

It remains only to ascertain the amount of this displace- 
ment. It depends on the relative velocities of light and of 
the earth. If light moved with infinite speed, there would 
be no aberration. Rays would leave the star and reach 
our eyes before we had changed our place. But its velo- 
city bears a definite ratio to the velocity of the earth in 
her orbit, 192000 : 19. To ascertain the angle of aberra- 
tion we should then construct a triangle, of which one side 
(the velocity of the ray) should be to the other (the velo- 
city of the earth), as 10105 : 1. The angle made by the 
ray and orbit being also known, we can find the other 
angles, and the direction of the third side of the triangle. 

§ 404. Let S (Fig. 5, Plate II.) represent a star. 
Let SB be the actual course of a ray. And, for simplicity, 
let S B be perpendicular to A B, the line of the earth's 
motion. The side A C represents the inclination which 
must be given to the eye, in order that when it has passed 
on to B, it may meet the ray from S. 

B C : B A : : vel. light : vel. earth : : rad. : tang. 20."246. 

Thus the angle B C A, or its equal S C E, by which the 
direction of the observer's eye deviates from the true direc- 
tion of the star=20."246. Also C B D, the displacement 
of the star, or the amount of aberratiom=20."246. 

Passing from A to B, the observer has gained upon the 
ray an angle A C B. Unconscious of his motion, he adds 
this angle to the inclination of the ray, as the man in the 
rain adds the angle made by his own motion to the true 
angle of the rain. 



ELEMENTS OF ASTRONOMY. 291 

The same reasoning holds good when the direction of 
the earth's motion is not perpendicular to the raj. The 
star may be so placed that S B A shall be an acute angle, 
or so that it shall be an obtuse angle. In either case we 
have the proportion : 

BC: BA: : sine of B AC : sine of AC B, orCBD, 
(the apparent displacement.) 

Thus it appears that the sine of the aberration is pro- 
portional to the sine of the angle made by the earth's mo- 
tion in space with the ray, and is therefore greatest when 
the line of sight is perpendicular to the orbit. 

§ 405. Aberration distorts the aspect of the heavens, 
causing all the stars to crowd as it were toward that point 
in the heavens which is the vanishing point of all lines 
parallel to that in which the earth is for the moment mov- 
ing. As the earth moves round the sun in the plane of the 
ecliptic, this point lies in that plane, 90° in advance of the 
earth's longitude, or 90° behind the sun's, and moves on- 
ward continually, describing the circumference of the eclip- 
tic in a year. It causes each particular star apparently to 
describe a small ellipse in the heavens, having for its 
centre the point in which the star would be seen if the 
earth were at rest. 

Let us consider the manner in which the phenomena of 
aberration succeed each other in the course of the year. 
Let A B a C, (Fig. 6, Plate II.), represent the celestial 
ecliptic, S the position of any star, and A S a the half of a 
great circle drawn through the star perpendicular to the 
plane of the ecliptic: and let B, C, be other points in the 
celestial ecliptic, and B S b, C S c, arcs of great circles 
drawn through the star and these points respectively, 
which will of course be semicircles, as all great circles bi- 
sect each other. If then A, B, C, a, b, c, represent differ- 
ent positions of the point toward which the earth moves, 
(or the point 90° before the earth's place), S A, SB, S C, 
S a, S b, S c, will represent the arcs, to the sines of which 
the amount of aberration is proportional, and in the direc- 
tion of which it takes place. 

§ 406. Now the earth being in every point of the 
ecliptic in the course of the year, every point of the celes- 



292 ELEMENTS OP ASTRONOMY. 

tial ecliptic must be 90° before its place in the course of 
the same period, and consequently the point S must be 
joined with every point of the celestial ecliptic to give all 
the arcs which determine the magnitude and direction of 
the aberration during the year. Of these the least is A S, 
the greatest S a ; the one being greater and the other less 
than 90° : and, in passing from A towards a, the corres- 
ponding arcs must pass through all intermediate values, in- 
creasing as they approach a. Of course, among these, 
there must be one which is of 90° ; let S C be this ; and 
the aberration in the direction of the line S C will have 
its greatest value, and will of course be 20. "246. The 
arcs C s, S c, together make a semicircle, and S C being 
90°, S c will be 90° also ; of course, therefore, the aberra- 
tion in the direction S c will also be 20. "246 ; and the ex- 
treme distance between the two apparent places as affected 
by the aberration in these opposite directions will be 
40. "492. If again, B S b represent any other great cir- 
cle, passing through S, the aberration in the direction SB 
will be proportional to the sine of S B, and that in the 
direction of S b will be proportional to the sine of S b. 
But the arcs SB, S b, together make up a semicircle, or 
S B is the supplement of S b : and as the sine of the arc 
and of its supplement are equal, the abberations in the 
directions SB, S b, are equal. In the same manner, the 
aberrations in the directions S A, S a, are equal ; and as 
S A is the least possible arc drawn from S to the celestial 
ecliptic, these are the least values of the aberration. The 
greatest values of the aberration are in the directions S C, 
S c ; and as the arcs S C, S c, are of 90° each, the circle 
CSc cuts the circle A S a at right angles ; or the direc- 
tions of the greatest and least aberrations are perpendicu- 
lar to each other, the least aberration taking place in a 
direction perpendicular to the ecliptic, or affecting only the 
latitude of the star. On investigation of the precise 
amount of the aberration in every direction, on the suppo- 
sition that it is the effect of the earth's motion, we shall 
find that the apparent place of the star is always in the 
periphery of the ellipse of which the centre is the true 
place of the star ; the minor axis is in the direction of a 



ELEMENTS OP ASTRONOMY. 293 

great circle passing through the star perpendicular to the 
ecliptic, the major axis=40."492 ; and the proportion of 
the minor to the major axis, that of sine of star's latitude to 
radius. Of course, therefore, the star is never seen in its 
true place, except in one case, which we shall presently 
mention. 

§ 407. It was said that aberration of light caused stars 
to move in an ellipse. This is true of all stars except 
those which are in the ecliptic, or of one in the pole of the 
ecliptic. Let us suppose the case of a star placed exactly 
in the pole of the ecliptic. In this case, the arc drawn 
from the star to every point in the ecliptic is exactly 90°, 
and its sine, therefore, always equal to the radius. Con- 
sequently the star will be seen in a curve parallel to the 
ecliptic ; it will always appear 20. "246 distant from its true 
place, and the amount of aberration will be always the 
same, and always the greatest possible. The star will al- 
ways be 90° further advanced in its orbit than the earth 
in its orbit. In the case of a star situated in the ecliptic, 
an arc drawn from the star toward the point of the ecliptic 
towards which the earth is moving, will always be a por- 
tion of the ecliptic itself. The whole aberration therefore 
will be in the plane of the ecliptic, or will take place en- 
tirely in longitude. The magnitude of this arc also will 
have every value from 0° to 180° ; therefore the aberration 
will, at two points where the arc is 90°, have its greatest 
value of 20. "246. And at other two, where the arc is 0° or 
180°, that is to say, when the earth is moving directly to- 
wards or away from the star, the aberration will be nothing. 
The star's latitude being nothing, the minor axis of the 
ellipse in which it is seen becomes nothing also, and the 
ellipse itself becomes a straight line, in which the star ap- 
pears to oscillate backwards and forwards ; passing through 
the centre, or having its apparent coincide with its true 
place in the course of its passage each way. All stars 
which are neither in the ecliptic nor the poles of the eclip- 
tic, describe ellipses of various forms, as described above. 

§ 408. The aberration we have now been considering, 
arises from the motion of the observer. We will now con- 
sider that which arises from the motion of the body ob- 
25* 



294 ELEMENTS OE ASTRONOMY. 

served ; its consequences are much more easily conceived 
of, and it affects only a few of the binary and multiple 
Stars whose motions have been watched. To distinguish 
the two kinds of aberration, the former has been called 
subjective, the latter objective aberration. Objective aber- 
ration then applies to those apparent displacements which 
originate in the length of time occupied in the transmission 
of light from the luminary to the eye. 

§ 409. Let us consider a Binary Star, consisting of 
two stars, A and B, at so great a distance that light re- 
quires x years to reach the earth. Suppose, for simplicity, 
the common proper motion of the centre of gravity to be 
in a direction perpendicular to the visual ray. It is ob- 
vious that each of the two stars A and B will be seen, in- 
dependent of the other at any given moment, not in the 
place which it occupies at that moment, but in that which 
it did occupy x years since, without regard to any change 
which may have taken place in its velocity or direction. 
Since this is true of each individual, it is true of both to- 
gether, regarded as forming a compound luminary, the 
parts of which must have, with respect to each other and 
to the spectator, an apparent situation identical with their 
real situation x years ago. We see, therefore, the com- 
pound object A and B in the state in which it really did 
exist x years previously. This being true at every in- 
stant, it follows that in viewing such a system continually 
for a series of years, we necessarily perceive its orbit in its 
true form ; and all the angles of position and distances in 
that orbit will be given truly by our measurements, unaf- 
fected by any optical illusion or distortion whatever, only 
for an epoch antecedent by x years. 

§ 410. There is a curious difference between the con- 
sequences of these two kinds of aberration, as shown in the 
heavens. Subjective aberration prevents our ever seeing a 
star at any moment in its proper place. We see it dis- 
placed by turns on every side. Objective aberration never 
removes a star from its true course, it merely causes it to 
lag behind and appear in a place historically, but not at the 
moment, true. 

The sun, moon, and planets suffer both subjective and 



ELEMENTS OF ASTRONOMY. 295 

objective aberration. Their subjective aberration may be 
found in the same way as that of the stars. Their objective 
aberration differs from that of the stars in this, that as we 
know their distances, we may ascertain how much they ap- 
pear to lag behind ; we may know how long since the place 
they now appear in was the true one. 

The sun being always in the celestial ecliptic, and at the 
distance of 180° from the earth, he is always 90° before the 
point toward which the earth is moving ; the aberration, 
therefore, in his case has always its greatest value, of 
20. "246. It always affects the longitude only, being in 
the plane of the ecliptic ; and it always diminishes the ap- 
parent longitude, because it brings the sun nearer the point 
toward which the earth is moving. We always, therefore, 
by the effect of aberration, imagine the sun to be in a point 
20. "246 behind the true direction of the ray by which we 
see him. 

The apparent places of the planets are affected in an 
analogous manner. The true place of any planet at the 
time of observation differs from the observed place by the 
arc which the planet describes in the time that a ray of 
light takes to pass from it to the earth. We must also 
allow for the motion of the earth during the same time. 

§ 411. In these calculations we have considered the 
earth's motion in its orbit only, and have placed the ob- 
server in the line joining the earth's centre and the sun. 
There must however be some aberration produced by the 
earth's rotation. The greatest possible velocity from this 
cause is that of a point in the equator, .2916 of a mile a 
second. The greatest aberration from this cause will be to 
that arising from revolution in proportion to their respec- 
tive velocities : 19 : .2916 : : 20."246 : .3108 of a second, 
a quantity too small to be regarded. 

We have also supposed the earth's motion in her orbit to 
be circular and uniform. The variations from these causes 
do not exceed 0."003. It would therefore be an unneces- 
sary refinement to allow for them. 



296 ELEMENTS OF ASTRONOMY* 



CHAPTER VIII 



TIME. 

Natural Divisions of Time. The Solar and Sidereal Day. Mean and 
Apparent Time. The Equation of Time. Variation in the Length of the 
Seasons. The Sidereal, Equinoctial and Anomalistic Years. Leap 
Year. Further Divisions of Time. 

§ 412. By seeing things change, and have a beginning 
and end, we acquire the idea of before and after. By 
equal intervals of time we mean successions of events dur- 
ing which we can execute the same things in the same 
manner, or during which the same phenomena are repro- 
duced in the same order. Unequal intervals are those in 
which we cannot do the same things, traverse the same 
road, or perform the same labor. It is then motion which 
gives the idea of time, and by motion time is measured ; 
for we cannot trust our senses to measure either. Time 
and motion are the only measures of each other. We can 
describe a time only by mentioning what portion of her 
daily or yearly course the earth has performed while it 
lasts ; we can describe the swiftness of a motion only by 
saying how much time it occupied. Thus the motion which 
we describe as occupying a given time must ultimately be 
referred to the time the earth occupies in another motion. 

§ 413. As we have described the motion of a body by 
saying that it moves a certain distance in a certain time, 
so we may now divide such portions of time as we can 
grasp into definite periods during which certain motions 
last. We cannot conceive of the beginning or end of time, 
but we can form a definite idea of periods of time. The 
most obvious periods are those of the revolution and the 
rotation of the earth, a year, and a day. But we find 
more than one kind of day, and more than one kind of 
year. A day, in common language, is the period between 
two successive appearances of the sun on the meridian, or 
between his rising and setting. A year is the period dur- 



ELEMENTS OF ASTRONOMY. 297 

ing which all the seasons return ; it may be reckoned from 
mid-winter to mid-winter, or from the vernal equinox to the 
vernal equinox, .or from a point at a certain distance from 
those points to the same again. But these divisions which 
are so obvious, and which regulate the labors of the hus- 
bandman and the common operations of life, are not nearly 
precise enough for astronomical purposes. The returns of 
the heavenly bodies to the meridian divide the year into 
convenient portions, the return of the sun to the same place 
in the heavens serves to mark when a larger portion of 
time has elapsed, but we must have some invariable 
standard of time, independent of the motions of the earth, 
with which to compare them. Such a standard we find in 
the oscillations of the pendulum. For the oscillations of a 
pendulum of a certain length, in a given latitude, must, in 
consequence of gravity, always occupy equal portions of 
time. Having this standard we may now ascertain whether 
these natural periods are always of the same length. 

§ 414. The two natural days are the solar and the 
sidereal. Of these the solar day is most convenient for 
common use, because every one knows how often in the 
year, and when, the sun is on the meridian, and his pre- 
sence there, and his rising and setting, control all our 
movements. But the sidereal day has this advantage, it is 
invariable. 

As the orbit of the earth is but a point compared to the 
fixed stars, the revolution of the earth does not change the 
length of a sidereal day. A place on the earth which is 
under a certain star at noon one day, is again under it in 
23h. 56m. 4" ; and returns to it after the same interval 
throughout the year. The different portions of each day 
are also described in proportional periods, as is shown by 
the apparent motion of the stars. If two stars are in the 
same circle of rotation, but one is 180° distant from the 
other, half of a sidereal day elapses between their appulses 
to- the meridian. If they are 90°, a quarter of aside- 
real day elapses ; and so on for every proportion of dis- 
tance. Therefore not only is the duration of a sidereal day 
constant, but during every part of it rotation goes on uni- 
formly. The pendulum also confirms these results. 



298 ELEMENTS OP ASTRONOMY. 

§ 415. A solar day is longer than a sidereal day. 
For while the earth has been once rotating, she has ad- 
vanced in her orbit nearly one degree, and this makes the 
sun appear to have advanced one degree. A place there- 
fore which has a certain star and the sun on its meridian 
one day at noon, must turn round further to bring the sun 
again on its meridian than to bring the star. Thus an 
absolute turn of the earth on its axis falls short of a 
natural day, and the earth requires as much more than 
one turn on its axis as it has gone forward in that time, 
which on an average is -g^b- part of a circle. Hence in 
365 days the earth turns 366 times on its axis. Thus 
there is one more sidereal day in the year than there are 
solar days of the earth or of any other planet ; one turn being 
lost by each planet's motion round the sun. From a simi- 
lar cause the traveller who journeys eastward round the 
world loses one day let him take what time he will. 

§ 416. We find by the pendulum that the solar days 
are not equal in length one to another, nor are they at all 
seasons of the year uniformly described. Yet the sidereal 
day will not answer to regulate the employments of life. 
For as the sun moving continually eastward is in the 
course of the year at all distances east of the star chosen 
to mark the noon of a sidereal day, if we reckoned by the 
star we should in the course of the year have noon when 
the sun was rising or setting, or at midnight. The sidereal 
day is therefore useless for common life, although it is 
employed by astronomers. 

We can, however, take the average of the solar days 
throughout the year, and thus obtain a mean solar day 
whose commencement never differs from that of the actual 
solar day much more than 16 minutes. The mean solar 
day is divided into 24 hours, each hour into 60 minutes, 
each minute into 60 seconds, and these are each of a fixed 
and determinate length. A pendulum 39.13929 inches, 
in the latitude of London, 51° 31' 1", in a vacuum at the 
level of the sea, vibrates seconds. The pendulum of an 
astronomical clock is usually made of such a length as to 
vibrate sidereal seconds. For the sidereal day may like- 
wise be divided into hours, minutes, and seconds. Compu- 



ELEMENTS OF ASTRONOMY. 299 

tations and observations made by sidereal time are easily 
transferred to solar time, and the reverse. 

§ 417. The common day begins at midnight, but the 
astronomical day begins at noon, and is counted from to 
24 hours. When it is noon, however, in one part of 
the world, it must be a different hour in all others not 
under the same meridian. Hence arises great inconven- 
ience, particularly as regards places situated widely apart 
in longitude ; the observed time must be referred to some 
common epoch. Time is therefore by astronomers referred 
to a fixed instant common to all the world, to the moment 
when the mean sun enters the mean vernal equinox. It is 
reckoned in mean solar days and parts of a day. Time 
thus reckoned is called equatorial time, and is numerically 
the same at the same instant in all parts of the globe. 

Sidereal time is calculated from the moment when the 
vernal equinox is on the meridian, and is also counted 
from to 24 hours. Clocks showing sidereal time are so 
regulated that they show Oh. ! 0", the instant the equi- 
noctial point passes the meridian of the observatory. And 
as time is a measure of angular motion, the clock gives the 
distances of the heavenly bodies from the equinox by ob- 
serving the instant at which each passes the meridian, and 
converting the interval into arcs at the rate of 15° an 
hour. 

§ 418. The variation in length of the solar day arises 
from two causes ; from the equator's not coinciding with 
the ecliptic, and from the unequal motion of the earth in 
its orbit. We will consider each of these causes inde- 
pendently of the other. The earth's motion on its axis 
being perfectly equable, and the plane of the equator be- 
ing perpendicular to its axis, it is evident that in equal 
times equal portions of the equinoctial pass the meridian ; 
and so might equal portions of the ecliptic, if the ecliptic 
were parallel to, or coincident with, the equinoctial. But 
as the ecliptic is oblique to the equinoctial, the equable 
motion of the earth carries unequal portions of the ecliptic 
over the meridian in equal times ; near the solstices smaller 
portions of the ecliptic rise in equal times than near the 
equinoxes. 



300 ELEMENTS OF ASTRONOMY. 

If on an artificial globe small patches are placed at 
every 15° of the equator and of the ecliptic, turning the 
globe slowly westward, all the patches from Aries to Can- 
cer, come to the brass meridian sooner than the corres- 
ponding patches on the equator. All those from Cancer 
to Libra will come later to the meridiem than their corres- 
ponding patches on the equator ; those from Libra to Ca- 
pricorn sooner, and those from Capricorn to Aries later. 
And the patches at the beginning of Aries, Cancer, Capri- 
corn, and Libra, will be either on or even with those on the 
equator. Thus from Aries to Cancer, the apparent time, 
as influenced by this cause only, will anticipate the mean 
time ; from Cancer to Libra mean time will be the fastest ; 
from Libra to Capricorn apparent time will again be faster, 
and from Capricorn to Aries it will be slower than mean 
time. If the ecliptic were more oblique to the equator 
than it is there would be still more variation. 

§ 419. The reduction of the apparent place of the sun 
at any time to its mean place is called the equation of time. 
Astronomical observations are presented not as they were 
observed but as they would have been observed if periodi- 
cal causes of fluctuation had not existed. Getting rid of 
these fluctuations is termed equating or correcting the ob- 
servation ; the amount added to or subtracted from the 
mean time is called the equation. 

§ 420. We will now consider the second cause of the 
inequality in the length of solar days ; this is the varying 
rapidity of the earth in different parts of her orbit. Near 
the aphelion her daily arc is no more that 57' 12", near 
the perihelion it is 1° 1' 19". From Aries to Libra her 
motion is more rapid than from Libra to Aries. At the 
aphelion apparent and equated time agree ; but the earth 
advancing from these with less than her mean speed, a 
place on her surface is brought under the sun in less than 
twenty-four hours, and apparent time is in advance of 
equated time shown by the clock. These differences of 
time increase until the increased speed of the earth begins 
to diminish them, and finally, at the perihelion, real over- 
takes apparent time. After passing the perihelion, the 
earth, moving with her utmost speed, gains upon the sun, 



ELEMENTS OF ASTRONOMY. 301 

and must make more than one rotation to bring the sun on 
the meridian of a given place. The difference between ap- 
parent and equated time increases a while, until diminished 
by the earth's slackened speed, and at the aphelion the 
times again coincide. Thus, owing to the excentricity of 
the earth's orbit, apparent time is in advance of mean time 
from mid-summer to mid-winter, and behind it from mid- 
winter to mid-summer. 

If the earth moved equably, but in the ecliptic, there 
would be no equation of time at the equinoxes and the sol- 
stices. If it moved with its elliptic irregularity, but in the 
equinoctial instead of the ecliptic, there would be no equa- 
tion of time at the aphelion and perihelion. Owing to these 
combined causes time presents a very irregular series of 
phenomena in the course of the year. The equation van- 
ishes only when the effect of one cause of irregularity is 
equal and opposite to the other. This takes place only 
four times a year ; on or near April 15th, June 15th, 
September 1st, and December 24th. As the perigee is 
in advance of the solstice, the two causes act partially to- 
gether in the autumn, while in the spring they partially 
counteract each other. 

Since the hours of daylight extend equally on each side 
of the apparent noon of each clay, the interval from sunrise 
to twelve o'clock must sometimes be longer and sometimes 
shorter than that from twelve o'clock to sunset. After the 
winter solstice of the northern hemisphere, when apparent 
time lags daily more and more behind the clock, and the 
hours of daylight are increasing, the sun rises at the same 
hour many days in succession, but sets later and later. 
Before the solstice, apparent time being in advance, and 
the differences between it and real time decreasing daily, 
the days appear to shorten in the morning only. 

§ 421. The results obtained above depend entirely on 
the relative positions of the aphelion and the equinoxes*. 
If these are fixed points, or always retain the same relative 
position, these results will serve alike for every year. If 
they vary, the equation of time will vary also. Any varia- 
tion in the inclination of the ecliptic to the equator would 
also affect it. In fact the inclination of the ecliptic to the 
26 



302 ELEMENTS OF ASTRONOMY. 

equator does undergo some slight variation, not important 
in its results, but furnishing one reason why calculations 
for the equation of time do not apply accurately except to 
the particular periods for which they were computed. The 
question whether the equinoxes and the aphelion are fixed 
points is more important, particularly in connection with 
the division of time into years. 

In seeking the relative position of the equinoxes and the 
perihelion we must remember that the equinoctial point 
is not stationary but slowly receding from east to west. 
The aphelion has also a slow motion of 11"66 eastward. 
The aphelion one year lies 11"66 east of the aphelion of 
the year before. And as this motion is in an opposite di- 
rection from that of the equinox, the difference between 
the equinox and the aphelion is constantly increasing by 
the amount of the two motions, or 61"9, annually. At 
this rate it would accomplish a revolution in the course of 
20,984 years. The major axis of the solar ellipse must 
have coincided with the line of equinoxes about 4,000 
years before the Christian era, much about the time which 
chronologists assign for the creation of man. In 6433 its 
major axis will again coincide with the line of the equi- 
noxes, but then the solar perigee will coincide with the 
autumnal equinox, whereas at the creation of man it coin- 
cided with the vernal equinox. 

§ 422. The variation in the position of the solar ellipse 
occasions corresponding changes in the length of the sea- 
sons. In its present position spring is shorter than sum- 
mer, and autumn longer than winter ; and while the solar 
perigee continues as it now is between the solstice of win- 
ter and the equinox of spring, the period including spring 
and summer will be longer than that including autumn and 
winter. In this century the difference is between seven 
and eight days. The intervals will be equal towards the 
year 6483 ; but when it passes that point the spring and 
summer taken together will be shorter than the period in- 
cluding autumn and winter. In the course of a revolution 
of the ellipse each season will by turns have been the 
longest and the shortest. 

This motion of the perigee and apogee, for of course it 



ELEMENTS OF ASTRONOMY. 303 

is common to both, is deduced from observation. If the 
very instant of the sun's being in apogee could be readily 
determined, this motion could easily be ascertained. For 
his place in the heavens might be determined for two suc- 
cessive returns to the apogee, and the difference of place 
would measure the motion. The variations of his apparent 
diameter or of his angular motion are however too slow 
to allow any very accurate estimation of very small dif- 
ferences. 

§ 423. The principle by which his apogee and perigee 
or the apsides of his orbit are found is very simple. We 
have seen that the radius vector describes equal areas in 
equal times. Now the only straight line which can be 
drawn through the focus of an ellipse so as to divide the 
ellipse into two equal parts is the transverse axis. If the 
sun be observed at any two points 180° distant from each 
other, he is at the two extremities of a line passing through 
the focus of the ellipse, and the portions of the ellipse on 
each side of that line must be unequal unless the line be 
the transverse axis, which passes through the apogee and 
perigee. If the portions are unequal his time of passing 
through them must be unequal also. If the times are 
equal, the instants of observation must have been when he 
was in apogee and perigee. If unequal, the place of the 
apogee and perigee may be found by calculation. The re- 
sult of observation is that the apsides have a progressive 
motion on the ecliptic of about 11. "8 annually. The lon- 
gitude of the perigee and apogee therefore increases at the 
rate of about 62" annually, 11. "8 by the onward motion 
of those points, and 50. "1 by the retrocession of the equi- 
nox from which it is measured. 

§ 424. We now see that there are three different pe- 
riods at which the sun may, in different senses, be said to 
return to the same position ; when he returns to the same 
equinox at which he was before ; when he returns to the 
same point in his orbit ; and when, having been in perigee 
or apogee, he returns to it again ; or, which is the same 
thing, when having been at a given distance from any of 
these points, he returns to the same point with respect to 
them. Each of these may be said to be the completion of 



804 ELEMENTS OF ASTRONOMY. 

a revolution of the sun ; and a revolution of the sun is 
called a year. The year from equinox to equinox is called 
the equinoctial year, or sometimes the tropical year ; for 
his time of returning from tropic to tropic, they being situ- 
ations always holding the same relation to the equinox for 
the time being, is obviously the same as that from equinox 
to equinox. The year from any point in the ecliptic to the 
same point again is called the sidereal year, for the sun is 
then in the same position as before, with relation to the 
stars. The sun's angular distance from the apogee is 
called the true anomaly, and the period between his leaving 
and returning to a given situation with respect to the apo- 
gee is therefore called the anomalistic year. 

§ 425. It is evident that the equinoctial is the shortest, 
and the anomalistic the longest of these years. When the 
sun starts from the equinox, it is a given point of his orbit ; 
before he returns to it, the equinox has receded on the 
ecliptic, and he therefore meets it again sooner than he re- 
turns to the same spot in his orbit. The effect therefore of 
the retrograde motion of the equinoctial point on the eclip- 
tic is to bring forward the time of the equinox, (or the in- 
stant at which the sun is at the equator) ; and hence, as 
we have already mentioned, the phenomenon is known by 
the name of the precession of the equinoxes. In the mean 
time, however, the apogee has moved forward on the eclip- 
tic ; and the sun, therefore, after returning to the same 
point in his orbit where he was at the former equinox, has 
still a further arc to describe before he arrives at his origi- 
nal position with respect to the apogee, and the time of his 
doing so is of course later. 

The mean length of the equinoctial year is 365d. 5h. 
48m. 51.6s., (or decimally 365d.242264) of mean solar 
time. After this, the sun has to describe 50. "1 to return 
to the same point of his orbit at which he was at the com- 
mencement of the year, or to complete the sidereal year ; 
and the mean length of the sidereal year is thus made 
365d ; 6h. 9m. 11.5s. or 365d.256383. He then has to 
describe a further arc of ll."8 to arrive at his original posi- 
tion with respect to the apogee, and the length of the ano- 
malistic year is thus made 365d. 6h. 13m. 58s. 8, or 
365d. 259708. 



ELEMENTS OF ASTRONOMY. 305 

§ 426. The lengths assigned to the equinoctial and 
sidereal years are only mean lengths ; that given to the 
anomalistic year is a true one. We shall hereafter shew, 
from other considerations, that the length of the anomalis- 
tic year does not vary. For the present, we will assume 
that fact, and then it is obvious that the length of the 
equinoctial and sidereal years must continually vary ; for 
each of these years is shorter than the anomalistic year by 
the time which the sun takes to describe a given angle of 
his orbit ; in one case 62", in the other 11. "8. Now the 
rate of the sun's motion is different in different parts of 
his orbit, faster as he is further from the apogee, slower as 
he approaches it ; and, consequently, his times of describ- 
ing these spaces of 62" and 11 ."8 continually vary as they 
are differently situated with respect to the apogee. The 
times therefore which are to be subtracted from the uni- 
form length of the anomalistic year, to ascertain those of 
the equinoctial and sidereal years respectively, are them- 
selves of variable duration ; and the lengths of the equinoc- 
tial and sidereal year are necessarily so too. The varia- 
tion, however, is very small, and the mean differs from 
the true length at any period by a very inconsiderable 
quantity. 

§ 427. It is obviously necessary, for many purposes, 
not only of chronology and history, but even of personal 
and domestic convenience, that we should have the means 
of dividing time into definite periods of considerable length ; 
and the most obvious and natural period to adopt, is that 
which includes all the various operations and appearances 
which succeed each other in regular order, which compre- 
hends seed-time and harvest, summer and winter. 

The tropical or civil year, of 365d. 5h. 48m. 49s. 7, is 
the time elapsed between the consecutive returns of the 
sun to the mean equinoxes or solstices, including all the 
changes of the seasons, is a natural cycle peculiarly suited 
for a measure of duration. It is estimated from the winter 
solstice, the middle of the long annual night under the 
north pole. But although the length of the civil year is 
pointed out by nature as a measure of long periods, the in- 
commensurability that exists between the length of the day 
26* 



306 ELEMENTS OF ASTRONOMY. 

and the revolution of the sun, renders it difficult to adjust 
the estimation of both in whole numbers. 

§ 428. If the revolution of the sun were accomplished 
in 365 days, all the years would be of precisely the same 
number of days, and would begin and end with the sun at 
the same point of the ecliptic. But as the sun's revolution 
includes the fraction of a day, a civil year and a revolution 
of the sun have not the same duration. Since the fraction 
is nearly the fourth of a day, in four years it is nearly 
equal to a revolution of the sun, so that the addition of a 
supernumerary day every fourth year nearly compensates 
the difference. But in process of time further correction 
will be necessary, because the fraction is less than the 
fourth of a day. In fact, if a bissextile be suppressed at 
the end of three out of four centuries, the year so deter- 
mined will only exceed the true year by an extremely 
small fraction of a day ; and if, in addition to this, a 
bissextile be suppressed every 4,000 years, the length of 
the year will be nearly equal to that given by observation. 
Were the fraction neglected, the beginning of the year 
would precede that of the tropical year, so that it would 
retrograde through the different seasons in a period of 
1,507 years. 

The division of the year into months is very old and 
almost universal. But the period of seven days, by far 
the most permanent division of time, and the most ancient 
monument of the common origin of the human race, was 
used by the Brahmins in India with the same denomina- 
tions employed by us, and was alike found in the calendars 
of the Jews, Egyptians, Arabs and Assyrians. It has sur- 
vived the fall of empires, and has existed among all suc- 
cessive generations. 



ELEMENTS OF ASTRONOMY. 307 



CHAPTER XIX 



PAEALLAX. 

Parallax defined. Horizontal Parallax. Methods of correcting for Parallax. 
Determination of the Moon's Parallax. Transits of Mercury and Venus. 
Methods of computing the Solar Parallax. Distances of the Sun and 
Planets. Parallax of the Fixed Stars. 

§ 429. We are now prepared to enter more particu- 
larly into the effects of parallax, both diurnal and annual, 
and to show the knowledge obtained by its means. 

The centre of the earth is really the place to which all 
motions in the solar system should be referred. The centre 
of the sun is the point to which all out of it should be re- 
ferred. Yet we are never in either of those places ; we 
are always 4,000 miles from the one, and upwards of 
95,000,000 from the other. Observations made on dif- 
ferent parts of the surface of the earth must therefore be 
corrected and referred to the centre of the earth. We 
must always know on what part of the earth, at what pe- 
riod of her rotation and of her revolution, a given observa- 
tion was made. 

But this eccentric position, which seems so disadvan- 
tageous, is the only foundation of an accurate knowledge 
of the absolute dimensions of the solar system. Without 
it we could not ascertain the distances, and consequently 
the real magnitudes of the heavenly bodies. 

§ 430. The sun, moon and planets assume different 
positions among the fixed stars which are at an incalculable 
distance, when viewed by two observers in different parts 
of the globe. This difference of position is termed the 
parallax of the heavenly body ; and when it is in the 
horizon of one of the observers such difference is called its 
horizontal parallax. 

To an observer at A, (Fig. 7, Plate II.) a heavenly 
body B, will appear in the horizon, either rising or setting, 
and he would refer its position among the fixed stars to the 



308 ELEMENTS OF ASTRONOMY. 

point G in a circle immeasurably distant, although the 
limits of the diagram compel us to contract its dimensions. 
Another observer at A', will have the same body, which we 
will suppose to be the moon, in the zenith ; a line joining 
C, the centre of the earth, and his position, will pass 
through B. This observer, then, views the object as it 
would appear from the earth's centre, and refers it to the 
point E ; a position distant from the former by the arc 
E G. Now if two observers remark the position of the 
moon at the same instant of time, and afterwards compare 
notes, the measure of this arc will be known, and conse- 
quently the value of the angle EBG, which is equal to the 
angle A B C, or the angle which the earth's semi-diameter 
would subtend when seen from B, the moon. 

§ 431. Since the triangle C A B, is right angled at A, 
we know A C, the earth's semi-diameter, the angles C B A 
and CAB; whence may be found the side C B, by the 
following proportion : 

Sine ABC: rad. : : C A : : C B. 

In the case of the moon, her mean horizontal parallax 
is bV 12" ; therefore, 

As sine 57' 12" : rad. : : 3,956 miles, the earth's semi- 
diameter : 237765, the moon's distance. 

Her diameter may be easily found when her distance is 
ascertained, (Fig. 8, Plate II.) : the angle E C G is that 
subtended by the diameter of the moon ; half of this will be 
E C B, which is one angle of the right-angled triangle 
E C B, of which the base C B is known, whence the per- 
pendicular E B may be easily found by trigonometry : this 
multiplied by two will give E G the diameter of the moon, 
which is about 2,000 miles : her mean distance is about 
240,000 miles. 

§ 432. As the heavenly body rises above the horizon, 
its parallax will become less and less ; thus, at H, (Fig. 7, 
Plate II.), its place, as seen from the centre, will be S ; 
from A, it will be W ; its arc of displacement S W being 
less than E G ; when at N, Q S will be less than S W ; 
while at M, in the zenith, its parallactic angle will vanish, 
and its position as seen from C, and also from A, will 
bel. 



ELEMENTS OP ASTRONOMY. 309 

When the parallax of a body for one altitude is known, 
its parallax for any other may be found from the following 
proportion : 

As racl. : sine of apparent zenith distance : : horizontal 
parallax : parallax in altitude. 

For in the triangle A C H, 

A C : C H : : sine A H C : sine CAH, or its supple- 
ment M A H. 

Again in triangle ABC, 

AC: CB: : sine ABC: sine C A B, or radius. 
And since the antecedents are equal, the consequents are 
proportional ; therefore, 

Sine A H C : sine M A H : : sine ABC: rad. 

Or, as rad. : sine M A H : : sine ABC: sine A H C. 

Again, A C being known, and C H found, as before ; in 
the triangle H A C given C H=C B ; the angle H A C= 
supplement of zenith distance, whence angle A H C, which 
is the parallax in altitude, may be found. 

The general effect of parallax causes heavenly bodies to 
appear nearer the horizon than their true place. Its true 
altitude is its observed altitude and its parallax. The hori- 
zontal parallax (Fig. 7, Plate II.) is less for a body more 
distant, as at D : where it is seen depressed only by the 
small arc F Gr. An exact proportion exists between the 
distances and horizontal parallaxes of two bodies : for in the 
triangle C B D, as sine B D C : sine CBD, or its supple- 
ment A B C : : C B : C D. That is, the distances of two 
bodies are in the inverse proportion of the sines of their 
horizontal parallaxes. 

§ 433. The two observers may be situated in different 
latitudes, but it is important that they should be under the 
same meridian. It is not likely that two observatories will 
be found situated under precisely the same meridian. 
Allowance must therefore be made for the change of the 
moon's actual zenith distance in the interval of time elaps- 
ing between its arrival on the meridians of the stations. 
Of course the nearer the stations are to each other in lon- 
gitude, the less is this interval of time, and the smaller the 
correction to be made. Suppose two observers, one in 
north and one in south latitude, to observe on the same day 
the meridian altitudes of the moon's centre. Having found 



310 



ELEMENTS OF ASTRONOMY. 



the zenith distances, and cleared them of the effects of 
refraction, if the distance of the moon were equal to that 
of the fixed stars, the sum of the zenith distances thus 
found would be precisely equal to the sum of the latitudes 
north and south of the places of observation. For it would 
be equal to the meridional distance of the stations across 
the equator. But the effect of parallax being in both cases 
to increase the apparent zenith distances, their observed 
sum will be greater than the sum of the latitudes, by the 
whole amount of the two parallaxes. This angle then is 
obtained by subtracting the sum of the latitudes from the 
sum of the zenith distances ; and this once determined, the 
horizontal parallax is easily found by dividing the angle so 
determined by the sum of the sines of the two latitudes. 
It is curious that these latitudes are originally learned only 
from comparison with the stars. Thus by taking a point 
on the earth, and comparing it with two others, one of 
which is infinitely distant and the other not so, we learn 
the distance of the latter. 

§ 434. The moon's parallax is as useful to us as her 
eclipses. They teach us the globular form of the earth 
and moon. The variation of her parallax for the same 
place at different times proves to us the eccentricity of her 
orbit. The variation at different parts of the earth, when 
she is at her mean distance from them in her orbit, shows 
the difference in the lengths of the terrestrial radii, and 
thus proves that the earth is not a perfect sphere. The 
equatorial parallax is ^fe greater than the polar parallax. 
This difference, though very small has a very considerable 
influence upon several astronomical phenomena ; for exam- 
ple upon the time of the occultation of a star by the moon, 
or even upon the occurrence of this phenomenon. 

Parallax acts only in a vertical circle ; it diminishes the 
altitude, and affects thus the right ascension, declination, 
latitude and longitude. This is of importance in many 
cases, and particularly with regard to solar eclipses. It in 
a measure counterbalances the effects of refraction ; it de- 
presses vertically, while refraction raises vertically. 

§ 435. Not only are the sun and moon displaced by 
parallax, but their apparent size is affected by it. Let us 



ELEMENTS OF ASTRONOMY. 311 

consider the distance of the moon from the centre of the 
earth as constant. It will then describe its daily circle 
round this centre, and will be nearer the observer when in 
the zenith than when in the horizon. If it is nearer, its 
apparent diameter will be greater. The increase, in pass- 
ing from the horizon to the zenith is about one sixtieth, be- 
cause in the passage from one of those positions to the 
other, the distance of the observer from the moon is dimin- 
ished by a quantity equal to the radius of the earth, which 
is about its sixtieth part. 

In the same manner, a cloud which near the horizon 
looked like a mere speck, when it drifts across our zenith 
appears a broad canopy, but subsides to its original size as 
it floats to a distance. 

Thus a small cloud interposed between us and the sun 
obscures it ; but an observer at a short distance refers the 
cloud to another place, and sees the sun also. Thus in a 
solar eclipse, the observer where the point of the shadow 
falls, loses sight of the sun entirely. To an observer on 
the east side of the earth, the east edge of the sun is ob- 
scured ; to one on the west side, the west edge is only ob- 
scured. As the planets and the sun are much more dis- 
tant from the earth than the moon is, it follows that their 
parallactic angles are much less than hers. 

§ 436. The sun's parallax indeed is so small that it 
cannot be obtained accurately by direct observation, but 
may be calculated from the parallax of a planet, or may be 
approximated in a rude way by the moon's parallax. On 
drawing a figure, it will be immediately seen that when the 
moon has completed her first quarter, the sun, the moon, 
and the spectator form a triangle which is right angled at 
the moon. Now the angle which separates the sun from 
the moon can be observed at the same instant ; suppose it 
== E, we have, % 

Distance earth and sun=distance earth and moon X sec. E. 

The exact moment when the moon is thus situated can- 
not be noted with accuracy, yet early observations made 
thus, showed that the sun was far more distant than the 
moon. 



312 ELEMENTS OF ASTRONOMY. 

Kepler's discoveries, that the planets move in ellipses 
round the sun in the focus, that the area swept by each 
radius vector in a given time is a constant quantity for the 
same planet, and, lastly, that the squares of the periodic 
times are as the cubes of the mean distances, have supplied 
means for a much more accurate determination of the sun's 
parallax. Assuming these laws, the forms of the orbits of 
the earth and planets, and their relative distances can be 
determined from observation ; hence, if the parallax of any 
one planet can be found, the parallaxes of the sun and of 
all the other planets can be computed. Observations of 
Mars, for instance, made at the Cape of Good Hope and 
at Greenwich, will afford a very tolerable value of his 
parallax, and hence of his distance. Then, as the propor- 
tion between the distances of Mars and the earth from the 
sun at any time is known from the form of their orbits and 
their periodic times, and the angle between the sun and 
Mars at the earth can be observed, the triangle between 
the sun, the earth, and Mars, can be solved, and hence the 
distance of the sun, and his parallax can be computed. 

§ 437. But the planets are so remote, and their paral- 
laxes consequently so small, that they cannot be ascer- 
tained with certainty. Their parallax would scarcely be 
perceptible except near the horizon, and there refraction 
prevents any great nicety of observation. 

It is only when an inferior planet passes between the 
earth and the sun, thaWery nice observations can be made 
both upon the planet and the sun. When Venus passes 
between the sun and earth, she appears like a dark round 
spot crossing the surface of the sun from the eastern to 
the western edge. Since the orbit of Venus can inter- 
sect that of the earth in only two points, the nodes, she 
is never seen upon the sun except when her conjunc- 
tions happen in or near the nodes of her orbit. At all 
other times she passes above or below the sun, and her 
dark side being toward the earth she is invisible. These 
transits take place at intervals of about eight and 113 
years. 

The transits of Mercury are much more frequent, but he 
is so much nearer to the sun than to the earth that his 



ELEMENTS OF ASTRONOMY. 313 

parallax is of little use in determining that of the sun. 
A small error in measuring would cause a much greater 
error in the snn's parallax. 

§ 438. Transits of Mercury and Yenus are interesting, 
as proving that they are nearer to the sun than the earth 
is, and that they shine only by reflected light. They also 
show the place of the planet's nodes. How often since 
the creation those planets had crossed the sun's disc we 
know not. It was not till 1769 that a transit of Venus 
was observed with sufficient accuracy to fix the horizontal 
parallax of the sun. For the sake of observing this transit 
many European governments sent expeditions to various 
parts of the world, and the result of their observations 
we now consider the mean parallax of the sun. To deter- 
mine the parallax with any degree of precision requires 
the best instruments, the most practised observers, and a 
combination of favorable circumstances. As a very minute 
error in measuring the parallactic angle at Venus would 
introduce an enormous error into the calculation of the dis- 
tance, it is not surprising that great anxiety should have 
been felt to determine this angle with the utmost exact- 
ness. 

§ 439. The chief fact to be borne in mind, in consid- 
ering a transit of Venus, is that the difference of the appa- 
rent beginnings and endings of the transit in different 
places on the earth's surface depends on the difference of 
the distances of the sun and Venus from the earth. If 
the planet were as far off as the sun, and really passed 
over the surface of the sun, there would be no sensible 
parallax in different parts of the earth. The nearer Venus 
is to the earth, the greater is her displacement to all ob- 
servers but those situated in a right line joining the centre 
of Venus and the centre of the earth. 

We must remember also that the apparent path of Venus 
depends partly on her motion, partly on the position and 
motion of the spectator. But as this is a complicated sub- 
ject we shall consider separately the several circumstances 
of the transit, and give several modes in which the parallax 
may be found. 

27 



314 ELEMENTS OF ASTRONOMY. 

§ 440. A right line passing through the centre of 
Venus and a given point of the earth and produced to 
the- sun's disc, will mark the path of Venus on the sun, 
as seen from the given point on the earth. 

When the given point is in one of the poles, there will 
be no parallax arising from longitude, because the pole 
does not move. But there will be a parallax from the 
latitude. The transit line seen from the pole will be 
parallel to that seen from the centre of the earth, though 
not coincident with it. When the given point is in any 
part of the surface whose latitude is less than 90°, 
there will be a parallax produced by the latitude of the 
place and also by its longitude. This latter parallax. will 
alter the transit line both in position and length; and 
will prevent its being parallel to the central transit line, 
unless when the axis of Venus's orbit and thafof the earth 
coincide, as seen from the sun ; and this may not happen 
in many ages. 

§ 441. Let us suppose the earth perfectly at rest, 
without rotation or revolution, and Venus moving from west 
to east, with the excess of her orbitual velocity over the 
earth's. That part of her orbit in which she would move 
during her transit on the sun may be considered as a 
straight line ; and therefore a plane may be conceived to 
pass through it and through the earth's centre. To every 
place on the earth's surface cut by this plane, Venus would 
be seen on the sun in the same path that she would de- 
scribe as seen from the earth's centre. And therefore she 
would have no parallax of latitude north or south ; but 
would have a greater or less parallax of longitude, as she 
is more or less distant from the meridian of each place 
during her transit. To all places above and below this 
plane she would have a parallax in latitude. 

The transit of 1769 passed over the southern hemisphere 
of the sun's disc. Therefore a northern parallax caused 
her to describe a longer line on the sun than when seen 
from the centre of the earth. A southern parallax caused 
her to describe a shorter line. 

To all places situated in a plane perpendicular to the 
orbit of Venus, prolonged to the centre of the earth, there 



ELEMENTS OF ASTKONOMY. 315 

will be no parallax in longitude. In all places eastward of 
this the parallax would be west of the sun ; to all places 
west of this the parallax would be east of the sun. 

§ 442. A favorable position for an observer is to be 
three hours west of the meridian which is in the plane of 
the earth's and the sun's centre and the pole of the orbit 
of Venus. After the transit has begun at the centre of 
the earth, it will be delayed at the western observer's place 
by almost the horizontal parallax of Venus. When the 
transit ends the observer will be three hour circles west of 
the meridian, and the end of the transit will be accelerated 
by nearly the horizontal parallax of Venus. If the observer 
is situated in high latitudes, being nearer the axis of the 
earth, the parallax from position will be less than at the 
equator. The parallax from rotation will be the same, for 
he moves with the same angular velocity. 

If now we suppose another observer standing over the 
pole, on the opposite meridian, in such a manner that while 
the former moves the opposite way from Venus, he moves 
the same way, it is evident that to him the duration of the 
transit will be longer than it would at the centre of the 
earth. He accompanies Venus in her orbit, and thus 
keeps her longer between him and the sun. His position 
would not be all the time in the illuminated hemisphere. 
He would see the beginning of the transit just before sun- 
set, and the end of it after sunrise. He must of course be 
so near the pole that his night lasts less than six hours, and 
in north or south declination according as the declination 
of the sun is north or south. Transits which take place in 
June are more favorable for observation than those in No- 
vember, because the northern hemisphere offers more points 
of observation. 

§ 443. I shall now give two modes of finding the 
parallax of Venus. 

In the first mode the observers are 90° apart. They 
each observe the time of the egress of Venus from the 
sun's disc. The time which elapses between these 
egresses is the measure of Venus's motion during this time. 
The arc passed through by Venus in this time is the 



316 ELEMENTS OF ASTRONOMY. 

difference between the parallax of the sun and^that of 
Venus. The parallax of Venus may be found by the 
second person's ascertaining her place at the moment 
when he first saw her leave the sun. The former arc 
may be subtracted from the latter, and gives the sun's 
parallax. 

Let D B A (Fig. 9, Plate II.), be the earth,. V Venus, 
and T t R the western limb of the sun. To an observer 
at B, the point t at that limb will be on the meridian, its 
place referred to the heavens will be at E, and Venus, as 
it leaves the sun's disc, will appear just within it at t. 
But, at the same instant, to an observer at A, Venus is 
west of the sun, in the straight line A V F ; the point t of 
the sun's limb appears at e in the heavens ; and if Venus 
were then visible, she would appear at F. The angle 
C V A is the horizontal parallax of Venus ; and is equal to 
the opposite angle F V E, whose measure is the arc F E. 
A t C is the sun's horizontal parallax, equal to the opposite 
angle e t E, whose measure is the arc e E ; and F A e, is 
the difference between the horizontal parallaxes of Venus 
and the sun, and is found by observing how much earlier 
in absolute time Venus's total egress from the sun is, as 
seen from A, than as seen from B, which is the time she 
takes to move from v to V in her orbit. 

§ 444. The motion of Venus at the time of her last 
transit, was known to be 4' of a degree on the sun's disc 
in 60' of time, or 4" in one minute of time. 

Let us suppose then that A (Fig. 9, Plate II.) is 90° 
east of B, so that when it is noon at B, it will be six in the 
evening at A ; that the egress, as seen from B, is at one 
minute before twelve ; but that, as seen from A, it is seven 
minutes, thirty seconds before six. Deduct six hours for 
the difference of meridians of A and B, and the remainder 
will be 6' 30" for the time by which the egress of Venus 
on the sun at t is earlier as seen from A than as seen from 
B ; which time being converted into parts of a degree is 
26", or the arc F e of Venus's horizontal parallax from the 
sun. For as one minute of time is to four seconds of a de- 
gree, so are 6J minutes of time to 26 seconds of a degree. 



ELEMENTS OF ASTRONOMY. 317 

§ 445. The horizontal parallaxes of the planets are 
inversely as their distances from the earth's centre. 

Therefore, if on the day of transit the horizontal parallax 
of Venus be ascertained, the sun's horizontal parallax, and 
consequently his distance from the earth, can be calcu- 
lated. 

The sun's diameter is previously known, and consequent- 
ly the distance from his centre of chords of any given 
length. Yenus displaced in opposite directions by parallax 
measures on the sun's disc two chords whose length is 
known from the duration of the transit in each place. The 
belt between them is twice the parallax. 

Let E (Fig. 10, Plate II.) be the earth, V Venus, and 
S the sun, and C I) the portion of Venus's relative orbit 
which she describes while crossing the sun's disc. Suppose 
A and B two spectators at opposite extremities of that 
diameter of the earth which is perpendicular to the eclip- 
tic, and to avoid complicating the case, let us omit all con- 
sideration of the earth's rotation, and suppose A and B to 
retain that situation during the whole time of the transit. 
Then, at any moment when the spectator at A sees the 
centre of Venus projected at a on the sun's disc, he at B 
will see it projected at b. If then the spectator could sud- 
denly transport himself from A to B, he would see Venus 
suddenly displaced on the disc from a to b ; and if he had 
any means of noting accurately the place of the points on 
the disc, he might ascertain the angular measure of a b 
as seen from the earth. 

§ 446. Though the distance between these points 
cannot be immediately ascertained, the breadth of the zone 
included between the two apparent paths of the centre of 
Venus across the sun's disc may be ascertained. Each 
observer need only, with the utmost care and accuracy, 
each at his own station, notice where it enters and where it 
quits the sun, and what segment of the sun's disc it cuts 
off. This can be done with great delicacy by noting the 
time occupied in the whole transit. For the relative angu- 
lar motion of Venus being precisely known, and her appa* 
rent path being very nearly a straight line, these times 
give a measure, on a very enlarged scale, of the lengths 
27* 



S18 ELEMENTS OF ASTRONOMY. 

of the chords of the segments cut off. And the sun's 
diameter being also known with great precision, the versed 
sines of these chords, and therefore their difference, or the 
breadth of the zone required, become known. To obtain 
these times correctly, each observer must ascertain the in- 
stants of ingress and egress of the centre. He must note 
the instant when the first visible impression on the edge 
of the disc at P (Fig. 10, Plate II.) is produced, or the 
first external contact ; and again, when the planet is just 
wholly immersed, and the broken edge of the disc just 
closes again at Q, or the first internal contact ; and he 
must make the same observations at the egress at R S. 
The mean of the external and internal contents gives the 
entry and egress of the planet's centre. The especial ex- 
cellence of this method consists in the nicety with which 
the first streak of light after the interior contact at ingress, 
and the last streak before the interior contact at egress, 
can be observed. 

§ 447. Though we have thus far been supposing the 
distances both of the planet and sun to be unknown, astron- 
omers have long been acquainted, from the theory of 
gravity, with the proportion they bear each other. For, 
by comparing the rapidity of the earth in its orbit with 
that of Venus, the power of the sun's attraction on each 
may be dedueed, and thence their relative distances and 
likewise the proportion their distances bear each other. 
The distance of Venus from the sun is to its distance from 
the earth as 68 to 27. Since A V a, B V b, (Fig. 10, 
Plate II.) are straight lines, and therefore make equal 
angles on each side V, ab will be to A B as the distance 
of Venus from the sun is to its distance from the earth ; 
as68:27::ab:AB; ab therefore occupies on the 
sun's disc a space 2J times as great as the earth's diame- 
ter, and its angular measure is about equal to 2J times the 
earth's apparent diameter at the distance of the sun, or 
which is the same thing, to five times the sun's horizontal 
parallax. Having the horizontal parallax, we may easily 
find the distance of the sun in miles. As sine of horizontal 
parallax 8.6 : rad. : : 3956 miles=the earth's radius : 
95,000,000=the sun's distance. 



ELEMENTS OF ASTRONOMY. 319 

§ 448. Knowing the distance of the earth from the 
sun, we may take other measurements in the system simply 
by observing angular distances. For instance, we may 
determine the distance of. an inferior planet by merely 
measuring its angular distance from the sun at the time of 
its greatest elongation. 

Thus if we measure the angular distance of Mercury 
from the sun at the time of its greatest elongation, and 
form a right-angled triangle by the lines joining the three 
bodies, we have given the distance of the earth from the 
sun, and the angle formed at the earth — whence may be 
found the distance of Mercury at the time of observation. 
This planet is found to vary its greatest elongation from 
the sun very considerably, this angle varying from 28° 48 f 
to 16° 12' ; whence we conclude that its orbit is very 
elliptical. The planet Venus, on the contrary, shows but a 
slight deviation from a circular path ; her angle of greatest 
elongation ranges between 47° 48' and 44° 57'. 

§ 449. The distance of a superior planet from the sun 
may be found by measuring the arc through which it ap- 
pears to retrograde, when in opposition to the sun. 

Let X (Fig. 11, Plate II.) be the place of Mars in op- 
position, and y that of the earth ; y v the arc described by 
the earth in a short period of time, for instance one clay ; 
% 3 the arc described by Mars in the same time. The 
periodic times being known, these arcs may be found sim- 
ply by dividing 360° by the number of days occupied in 
one revolution. Since the earth's motion is more rapid, it 
will pass by Mars, and Mars will appear to retrograde 
through the arc rd, whereas in fact he has been moving 
through r d'. Now in the triangle S v 3, the angle S /3 v 
the difference between the heliocentric and geocentric 
places of Mars ; the angle 3 S v, which is the difference 
between the angular advance of the earth and that of 
Mars for a day ; and the side S v are given ; whence 
may be found by trigonometry S 3, which is the distance 
of Mars from the sun. 

§ 450. The whole parallax at a given distance is al- 
ways equal to the angle under which the earth's diameter 
is viewed from that place. The more distant the observer ? 



320 ELEMENTS Otf ASTRONOMY. 

the smaller is the angle. All bodies within the solar sys- 
tem have some parallax to one another. An inhabitant of 
Jupiter would have a more perceptible parallax for bodies 
at the same distance than an inhabitant of the earth, be- 
cause Jupiter would subtend to such a body a larger angle 
than the earth. In annual parallax Jupiter, and the outer 
planets have, from the superior size of their orbits, a still 
greater advantage. Perhaps to them the parallax of the 
fixed stars is distinctly visible. But if the inhabitant of 
the earth would go beyond this system, and would displace 
the stars, he must seek no longer for a diurnal but an 
annual parallax ; he must substitute 190,000,000 for 8,000 
miles, and reduce observations to the centre of the orbit, 
not that of the earth. When the Copernican theory of 
placing the sun in the centre was first introduced, a strong 
objection to it was, the enormous distance at which it re- 
quired the fixed stars to be placed. It was argued that 
the earth in her yearly circuit must displace them, and 
cause them to perform orbits parallel to the ecliptic round 
their true place, similar and equal to the earth's orbit 
round the sun. And further, that as parallax is always in 
a plane passing through star, sun and earth, it would di- 
minish the angle which the sun and star subtend at the 
earth. But the angles remained unaltered, the circles 
were not described, the stars were compelled to take a 
position so distant that the whole orbit of the earth was 
from them an invisible point. 

§ 451. When the motion of the earth in an orbit was 
first discovered, Galileo predicted that annual parallax 
would reveal the distance of the fixed stars. But not the 
closest investigation, renewed from time to time as instru- 
ments of greater power inspired new hope, resulted in any 
thing but a conviction that the parallax, if any existed, was 
inconceivably small. The delicacy of the researches made 
may be judged from the discoveries elicited in the course 
of them. While searching for parallax, Bradley discov- 
ered nutation, and aberration of light. Sir William Her- 
schel learned the connection and orbitual motion of the 
Binary Stars. But the question of the distance of the 
fixed stars remained unanswered for three centuries. It 



ELEMENTS OP ASTRONOMY. 321 

has lately been solved by means of more delicate instru- 
ments and a better choice of stars. There are only two 
circumstances to guide in the selection of stars for observa- 
tion, their remarkable brilliancy or greater apparent mo- 
tion. Both seem to indicate the comparative nearness of 
the star ; and of course the nearer the star the greater the 
parallax. 

§ 452. Sir William Herschel first pointed out the 
mode of detecting parallax, which has since proved suc- 
cessful ; namely, that which depends on the measurement 
of double stars. If the stars which compose a double star 
be at different distances from the earth, they must be dif- 
ferently affected by parallax, and therefore their apparent 
distance from each other will be altered by a change of 
position in the spectator. The apparent distance of two 
neighboring stars can be measured with great accuracy. 
It is more easy to determine the distance of a double star 
to 0. ,; 1, than to fix an absolute place to 1", The problem 
of parallax is therefore reduced to that of finding a double 
star in which a variation of distance is observable, and fol- 
lowing the law which the earth's change of place requires. 
It was during this inquiry that Sir W. Herschel discovered 
that very many double stars have a relative motion, both 
in distance and in angular position, which proves them to 
be a connected system. Sir John Herschel showed that 
the variation produced by parallax in the angle of position 
of two stars is a more sensible phenomenon, and one more 
easily measured than the variation in distance, and he pub- 
lished a list of stars suitable for this research, with the 
times of year when the observations would show the great- 
est effect of parallax. 

§ 453. It has been shewn that the earth's change of 
place in its orbit would cause each star to have an appa- 
rent motion in an ellipse of which the major axis is parallel 
to the ecliptic and equal to the diameter of the earth's 
orbit, as seen at the distance of the star. If then the 
stars be a connected system, and comparatively near each 
other, the stars will appear to describe two equal and simi- 
lar ellipses, and the line joining their apparent places will 
be equal and parallel to the line joining their true posi- 



822 ELEMENTS OP ASTRONOMY. 

tions. It is therefore in vain to look for any effect of par- 
allax in microinetrical or relative measures of distance and 
position in a connected system. But if one of the stars be 
much farther from us than the other, suppose it ten times 
farther off, then the apparent ellipses will continue to be 
similar, and similarly described ; only the dimensions of 
that described by the more distant star will be one tenth 
of that described by the nearer star. Or we may suppose 
the more distant star to be fixed, and the nearer star to 
describe round its true place an ellipse of nine-tenths the 
actual dimensions. 

§ 454. The apparent places of the stars being simi- 
larly situated in the two ellipses, their apparent distance 
on the line joining these apparent places will both oscillate 
in angular position and fluctuate in length, thus causing an 
annual relative alternate movement between the stars both 
in position and distance which is greater, the greater the 
difference between the parallaxes. Thus it is not the abso- 
lute parallax of either, but the differences of their paral- 
laxes, which is measured by this method. But when the 
stars are very unequal in magnitude, it is probable that the 
difference of their parallaxes very nearly equals the whole 
parallax of the nearer one. 

Most of the close double stars are probably connected, 
and therefore unfit for the detection of parallax by com- 
parison with one another. Yet the distance must not ex- 
ceed a few seconds, for the eye, in delicate measurement, 
must see both stars at the same glance. The number of 
stars apparently double, yet so close as to admit accurate 
measurement, is probably small. 

§ 455. The selection of a star for observation involves 
many considerations. The one chosen by Mr. Bessel, 
61 Cygni, is a fine double star ; and one that has been 
ascertained to be physically double. The distance of its 
individuals is great, being about 16J". This being neces- 
sarily less than the axis of their mutual orbit, affords in 
itself a presumption that the star is a near one. And this 
presumption is increased by the unusually great proper 
motion of this binary system, which amounts to nearly 5" 
per annum, and which is not shared by several small sur- 



ELEMENTS OF ASTRONOMY. o23 

rounding stars. Moreover, the angular. rotation of the two 
one about another has been well ascertained. Of course 
the proper motion and that of rotation were both allowed 
for in the calculation of the parallax of the central point. 
Of these small surrounding stars, two are very advanta- 
geously situated for comparison with either of the individ- 
uals of the binary star, or with the middle point between 
them. One of these (a), at a distance of 7' 42", is situated 
nearly at right angles to the direction of the double star ; 
the other (b), at a distance of 11' 46", nearly in that di- 
rection. Considering a and b as fixed points, and mea- 
suring at any instant of time their distances from c, the 
middle point of the double star, the situation of c relative 
to a and b is ascertained. And if this be done at every 
instant, the relative place of c, or the curve described by 
it on the plane of the heavens with respect to the fixed 
base line a b, will become known. 

§ 456. Now on the hypothesis of parallax, this curve 
ought to be an ellipse of one certain calculable eccentricity, 
and no other. And its major and minor axes ought to hold with 
respect to the points, a, b, certain calculable positions and 
no others. The distances a c and b c will each of them be 
subject to annual increase and diminution, in a given calcu- 
lable ratio the one to the other ; and the maximum and 
minimum of the one distance a c will be nearly contempo- 
raneous with the mean values of the other distance b c, and 
vice versa. 

Thus we have in the first place several particulars inde- 
pendent of mere numerical magnitudes ; and in the second 
place, several distinct relations a priori determined, to which 
those numerical values must conform, if it be true that any 
observed fluctuations in their distances ab and ac are 
really parallactic. So that if they are found in such con- 
formity, and the above mentioned maximum and minimum 
do observe that interchangeable law above stated, there is 
accumulated a body of evidence in favor of the resulting 
parallax, to which no reasonable mind can refuse its belief. 
Mr. Bessel found that the particulars observed agreed with 
those calculated in a signal and satisfactory manner. 



324 ELEMENTS OF ASTRONOMY. 

§ 457. The distances were observed for several months, 
and it was found that the distance between one star and the 
central point diminished, while that between the other and 
the central point increased. The observations were cor- 
rected for the proper motion, for aberration, and even an 
allowance was made for the effect of the temperature on 
the micrometer screw. These being allowed for the dis- 
tances obtained on the successive nights of observation, 
should have been exactly equal. This, however, was not 
the case. The observed and computed differences differed, 
and those differences seemed to follow a certain law, and it 
soon became manifest that the errors could be greatly re- 
duced by admitting a certain amount of parallactic motion 
of the central point of the components of the double star. 
The parallax obtained thus was 0."3309, which gives us 
the distance of the binary system of stars from our sun 
670,000 X 95,000,000 of miles; a distance so great that light 
can traverse the interval only after a flight of nine years 
and three quarters, at the velocity of twelve millions of 
miles a minute. The diameter of the orbit described by the 
two stars of 61 Cygni round each other is about 50 times 
that of the earth's orbit, or 2J times that of Uranus. 
Their periodic time is about 540 years. Such is the uni- 
verse in which we exist, and which we have at length 
found the means to subject to measurement. 

§ 458. Once having learned the distance of a fixed 
star, if it were possible to measure its apparent diameter, it 
would be easy to ascertain its magnitude. But the fixed 
stars show no discs even with the highest powers. They 
increase in brilliancy so as to be painful to the eyes, and 
are surrounded by a haze or dawn like that of morning, but 
they remain mere points. A telescope which brings them 
two thousand times as near fails to give them any sensible 
diameter. 

After years of incessant labor the parallax of the great 
star a Centauri, the third star of the heavens in brightness, 
has lately been ascertained to be a little more than nine 
tenths of a second, indicating a distance so enormous, 
that if our sun were so large as to fill the whole orbit of 
the earth, the sun when seen through a powerful telescope 



ELEMENTS OF ASTRONOMY. 325 

would have a radius of only nine tenths of a second. 
It is a magnificent double star ; one star being of a deep 
and brownish orange, the other of a fine yellow color ; 
each star being of the first magnitude. Their distance 
is at present (1850) about 9", but it is rapidly diminishing, 
and in no great lapse of time, they will probably occult one 
another, their angular motion being comparatively small. 
Their apparent distance was formerly much greater, how 
much we cannot say for want of observations, but proba- 
bly the major axis of their mutual orbit is short of a 
minute of space. They therefore afibrd strong indications 
of being very near our system. Added to which their proper 
motion is very considerable, and participated in by both, 
which proves their connection as a binary system. An 
additional presumption in favor of their proximity may 
be drawn from their situation in the nearest region of the 
milky way, among a great number of large stars. 

The parallax of the double star a Lyrae has also been 
found by M. Struve of Dorpat. Its distance from the 
solar system is 771,400 radii of the earth's orbit, a space 
passed through by light in twelve years. 



CHAPTER XX. 

THE SUN AND PLANETS. 

The Mass and Dimensions of the Sun. Its Atmosphere. Its light and 
Heat. The Solar Spots. Proposed explanations of their Appearance. 
The Sun's Rotation. The Centre of Gravity of the Solar System. 
Determination of the Orbits, Masses and Densities of the Planets. 

§ 459. The sun always presents a round luminous 
disc, and since the telescope has shown its rotation, this is 
a proof that its form is nearly or quite spherical. It is the 
only fixed star so near that we can learn its diameter and 
the appearance of its surface. Its mass is 800 times that 
of all the planets taken together. It is only 329,000 times 
28 



326 ELEMENTS OF ASTRONOMY. 

that of the earth, though its solid contents are to those of 
the earth as 1,300,000 to 1. Hence it attracts the earth 
only 329,600 times as forcibly as the earth attracts it. 
In speaking of its size, we include the luminous atmo- 
spheres, which are at least several thousand miles deep. 
The solid nucleus of the sun may have a much greater spe- 
cific gravity than has here been assigned to the whole orb. 
Its diameter is 111 times that of the earth. Its actual diam- 
eter is 900,000 miles. If its centre were to coincide with 
the centre of the earth, its volume would not only include 
the orbit of the moon but would extend nearly as far again. 
It could contain within its circumference more than 130,000 
globes as large as the earth, and a thousand globes as large 
as Jupiter. Its surface seems to have a state of constant 
ebullition and commotion or rather instability. Agitated 
apparently by powerful causes, it often accumulates into 
masses like waves, whose summits now round, now in 
ridges, constitute bright places, and which seen through a 
telescope, are finely mottled, and far from uniformly bright, 
resembling a mackerel sky. 

§ 460. From the properties of the sun's rays it is 
known that this incandescent substance, which causes light 
and heat, is neither a solid nor a liquid, but a gas. The 
intensity of the sun's heat is proved by the facility with 
which the calorific rays traverse glass, a property which is 
found to belong to artificial fires in proportion to their in- 
tensity. The most vivid flames disappear, the most in- 
tensely ignited solids appear only as black spots on the disc 
of the sun when held between it and the eye. From this 
last fact it follows that the body of the sun, however dark 
it may appear when seen through its spots, may be in a 
state of complete ignition. Light and heat may arise from 
chemical changes taking place at the surface, from elec- 
tricity, or from some cause unknown to us. The sun's di- 
rect light has been estimated to be equal to 5,563 wax 
candles placed at the distance of one foot from the eye. 
That of the moon is only equal to one candle placed at a 
distance of twelve feet. 

§ 461. Its heat and light are sent forth equally in all 
directions. When a planet is turned toward it, we become 



ELEMENTS OF ASTRONOMY. 327 

aware of these rays by their reflection ; but they are as 
numerous throughout the whole system. We know its heat 
only by that intercepted by the earth, yet of that alone as 
much as is received in one year is sufficient to melt a 
stratum of ice forty-six feet deep covering the whole globe. 
Part of this heat is radiated back into space, but by far the 
greater part descends into the earth during summer, and 
returns thence in the course of the winter. The fixed stars 
are too distant to afford us sensible heat. The heat received 
on a given area exposed at the distance of the earth, and 
on an equal area at the visible surface of the sun, must be 
in the proportion of the area of the sky occupied by the 
sun's apparent disc to the whole hemisphere, or as 1 to 
about 300-,000. A far less intensity of solar radiation col- 
lected in the focus of a burning glass suffices to dissipate 
gold and platina in vapor. The planets and the moon, 
though so near, shining only by reflected light, give no 
sensible heat, though chemical rays have been detected in 
moonlight. It would require 90,000 moons, as many as 
would fill the whole of our visible sky, to afford us light 
equal to that which we have in a cloudy day when the sun 
does not shine out. 

§ 462. It is now believed that the sun consists of 
a solid nucleus enveloped by an elastic non-luminous 
atmosphere, supporting luminous strata which to us ap- 
pear as the disc of the sun. We infer the existence 
of this atmosphere from the appearances which accom- 
pany the spots seen in the sun. These spots are per- 
fectly and intensely black, but are surrounded by a penum- 
bra of a nearly uniform half shadow, thus presenting the 
appearance of a deep pit. There is no gradual melting of 
one shadow into the other, of spot into penumbra, penum- 
bra into full light. The idea conveyed is more that of the 
successive withdrawal of veils, the partial removal of defi- 
nite films, than of the melting away of a mist, or the 
mutual diffusion of gaseous media. The only theory which 
at all accounts for them supposes the black part of the spot 
to be the opaque body of the sun laid bare, or at least an 
opening, and the surrounding penumbra to be the non- 
luminous atmosphere. Luminous matter often is seen piled 



328 ELEMENTS OF ASTRONOMY. 

up round the edgos of the spot, and often precedes the 
breaking out of a spot, indicating great agitation of the 
atmosphere. 

The immense scale on which these spots take place, and 
the rapidity of their changes, confirm the idea that they 
take place in a mobile fluctuating gaseous atmosphere. 
A single second of angular measure seen from the earth, 
corresponds on the sun's disc to 465 miles. And a circle 
of this diameter (containing therefore nearly 220,000 
square miles) is the least space which can be distinctly 
discerned on the sun as a visible area. A spot seen on the 
29th of March, 1837, occupied an area of nearly five 
square minutes, equal to 3,780,000,000 square miles. 
The black centre of the spot, May 25, 1837, (not the 
tenth part of the preceding one), would have allowed the 
globe of the earth to drop through it, leaving a thousand 
miles clear of contact on each side of the tremendous gulf. 
That such a spot should close up in six weeks time, its bor- 
ders must approach at the rate of more than 1,000 miles a 
day. 50 and even 150 spots have been seen at once on 
the sun. Sometimes several unite, and often a large one 
divides into several smaller ones, which soon vanish. They 
frequently appear in clusters, as at the time of the annular 
eclipse of the sun in 1836, when there were five near to- 
gether. 

§ 463. These spots have a motion of their own inde- 
pendent of the sun's motion. These two facts have sug- 
gested an explanation of them from analogous phenomena 
which take place in the earth's atmosphere. The spots on 
the north side of the equator, move northward, and either 
join and disappear, or burst, one large one forming several 
small ones. Spots on the south side move southward, 
and exhibit the same changes. 

On the earth hurricanes occur within the same latitudes, 
never cross the equator, and either gradually subside, or 
burst into smaller storms, and are dissipated. Hurri- 
canes are moving cylinders of air which rotate with ex- 
treme rapidity, and are slowly translated, thus resembling 
the spots on the sun which have vorticose movement. 
These motions when left to themselves die away, the lower 



ELEMENTS OF ASTRONOMY. 329 

portions coming to rest much more speedily than the upper, 
by reason of the greater resistance below, and of the chief 
action's being above, so that their centre, like that of our 
water-spouts, appears to retreat upwards. In the same 
way the solar spots appear to fall in by the collapsing of 
their sides, the penumbra closing in upon the spot, and 
disappearing after it. 

The spots also follow one another in lines parallel to the 
sun's equator, so that they are undoubtedly connected with 
the sun's rotation. It only remains therefore to inquire 
how such a circulation can be caused in the sun, so far as 
we know and understand it. It is evident this circulation 
on the earth depends on one part of the earth's becoming 
more heated than others, by exposure to the external source 
of heat ; but the same effects would be produced if the 
heat of the sun's surface were equal in every part and 
escaped unequally. 

§ 464. Sir John Herschel finds this cause of inequality 
in the thickness of that atmosphere of the sun which ex- 
tends beyond the luminous portion. He says that it has 
long been a question with astronomers, whether any such 
atmosphere exists. He considers the question settled in 
the affirmative by the rose-colored solar clouds seen in the 
total eclipse of July 8, 1842, which must have floated in 
and been sustained by an extensive transparent atmosphere. 
The deficiency of light at the borders of the sun's disc can 
only be caused by an atmosphere. To what distance this 
atmosphere may extend we have no means of judging accu- 
rately ; but from the manner in which the dimness comes on, 
being by no means sudden on approaching the edge, but 
extending to some distance within the disc, it would seem 
to be considerable, not merely in absolute measure, but as 
an aliquot part of the sun's radius. The equatorial and 
polar portions of this envelope may differ in density, and 
thus oppose unequally the escape of the sun's heat. 

- § 465. Those spots which remain stationary on the 
sun's surface for a considerable time have a gradual motion 
across the sun's disc, making their appearance on the 
eastern edge of the sun, in about a fortnight, if they con- 
tinue visible so long ; they move toward the western edge 
28* 



830 ELEMENTS OE ASTRONOMY. 

and disappear behind it. A fortnight after, the same spot 
sometimes reappears on the eastern edge. 

This motion of spots can only arise from the rotation of the 
sun and serve to mark its time of rotation. If the earth 
were stationary, this time would be equal to the time which 
elapses between two consecutive appearances of the spot 
on the western edge. But a correction must be introduced 
into the calculation, in consequence of the earth's motion, 
which, in the mean time, has been going on in the same 
direction as the motion of the sun on its axis. Thus, sup- 
pose the earth to be at E (Fig. 12, Plate II.) when a spot 
disappears ; if the earth stood still at E, the inhabitants 
would again see the spot in the same place after one revo- 
lution of the sun on its axis ; that is, when the spot had 
again arrived at b. 

§ 466. But in the mean time, the earth has advanced 
to D ; the spot has therefore to describe the additional 
arc a b before it will disappear. Now the arc a b, 
which measures the angle a c b, is equal to the arc D E, 
which measures the angle D a E, or the portion of the 
earth's orbit which she has passed over in that time ; 
hence, as 360°-f-ab, the whole space described by the 
spot : the whole time elapsed between the twc#disappear- 
ences : : 360° : the true time of the revolution of the sun 
on its axis, as it would have appeared to the earth had it 
been stationary at E. This time has been found to be 
25.01154 sidereal days ; two days less than it appears, 
owing to the earth's motion in the same direction. 

These spots also prove that the sun is a spherical body ; 
for a spot makes its appearance on the edge of the sun 
as a line, which gradually increases in breadth as it 
approaches the centre. As it passes on to the eastern 
edge, its diameter gradually lessens into a line before 
it entirely vanishes from view. Such appearances could 
arise only from the rotation of a spherical body. 

§ 467. The sun's spots not only prove his rotation and 
time of rotation, but they give us the plane of it, and show 
us where to draw his equator. 

If the axis of rotation were perpendicular to the ecliptic, 
the movement of the spots would always be rectilinear, and 
parallel to the line which the ecliptic marks on the sun's disc. 



ELEMENTS OF ASTRONOMY. 331 

But there are only two seasons of the year when this hap- 
pens, in February and in August. Soon after this the 
motion of the spots becomes curvilinear, and three months 
after like an arc, which would have for a cord a parallel to 
the ecliptic. At the end of May, the convexity of this arc 
is toward the south ; at the end of November it is toward 
the north. These phenomena prove that the sun's axis is 
inclined to the plane of the ecliptic. If this axis is so situ- 
ated that at the end of February and of August it is at the 
edge of the visible disc of the sun, the eye of the terrestrial 
spectator will be in the plane of the sun's equator pro- 
longed, and the path of the spots will be rectilinear. But 
three months after, the sun's pole will either be elevated 
above or sunk below its former position, so that all its lines 
of latitude will appear curvilinear. This axis is inclined 
to the ecliptic 7° 20'. It is directed towards a point half 
way between the polar star and Lyra, the plane of rotation 
being inclined a little more than 7° to that in which the 
earth revolves. 

§ 468. Besides rotation, the sun has a slight, irregular 
motion of revolution performed round the centre of gravity 
common to him and all his planets. The point to which the 
planets gravitate is not the centre of the sun, but the centre 
of gravity of the system ; the sun and the planets always 
balance one another round this point. This causes the sun 
to describe an orbit about the centre of gravity of the sys- 
tem, which is a very complicated curve, because it results 
from the action of a system of bodies perpetually changing 
their relative positions. If all the planets were in a straight 
line with the sun, and on the same side with him, which, 
however, from the periods of the planets never can happen, 
the centre of the sun would be as remote as possible from 
the common centre of gravity. Yet this distance would 
be not more than .0085 of the radius vector of the earth. 
We may form a better idea of the magnitude of this orbit 
by comparing it with that of the moon. A body revolving 
round the sun in contact with his surface, must be nearly 
twice as remote from his centre of gravity as the moon is 
from the earth, and the sun in his revolution round the com- 
mon centre of gravity of the system, must, when most re- 



332 ELEMENTS OF ASTRONOMY. 

mote, be four times the distance of the moon from the 
earth. 

§ 469. Before speaking of the physical constitution of 
the planets, we shall introduce a few remarks on the nature 
of the orbits which they describe, and of the methods of 
ascertaining their masses and density. For the complete 
determination of elliptic motion, the nature and position of 
their orbits must be observed. This depends on six 
quantities called the elements of the orbit, the modes of ob- 
taining which, when not learned by direct observation, have 
been already shown. These are, the length of the major 
axis, and the eccentricity, which determine the form of the 
orbit ; the longitude of the perihelion ; the inclination of 
the orbit to the plane of the ecliptic, and the longitude of 
its ascending node. These give the position of the orbit in 
space ; but the periodic time, and the longitude of the 
planet at a given instant, called the longitude of the epoch, 
are necessary for finding the place of the body in its orbit 
at all times. 

The only one of these quantities which is invaria- 
ble in each orbit is the length of the major axis, which 
equals the earth's greatest and least distance from the sun. 
The variations of the other elements are explained in the 
proper connection. 

§ 470. The periodic time of a planet is the interval 
between its quitting a node and returning to it again. Its 
nodes are the points where its orbit intersects the ecliptic. 
The ascending node is that point in the ecliptic through 
which the planet passes in going from the southern to the 
northern hemisphere. The descending node is a point in 
the plane of the ecliptic diametrically opposite to the former, 
through which the planet descends in passing from the north- 
ern to the southern hemisphere. 

The passage of a planet through its node is seen when it 
actually occurs, and in its true place, whether the planet's 
motion at the moment appear to be slow or swift, direct or 
retrograde. It is easy to ascertain by observation the time 
when a planet crosses the ecliptic. Its right ascensions 
and declinations are converted into longitudes and latitudes, 
and the change from north to south latitude shows in what 



ELEMENTS OF ASTRONOMY. 333 

day the transition took place ; while a simple proportion, 
grounded on the observed rate of its motion in latitude in 
the interval, suffices to fix the precise hour and minute of 
its arrival on the ecliptic. Suppose, by two observations 
24 hours apart, it had been found to be 3° N. L., and 5° 
S. L. As 8° : 24h. : : 3° : 9h. : : 5° : 15h. It crosses the 
ecliptic at nine o'clock. This having been done for several 
transitions, and the dates having been thereby fixed, the 
interval of time is found to be always the same. This pe- 
riodic time is the same, whether the body moves in a cir- 
cular or elliptic orbit, provided only that the mean dis- 
tance or half the major axis of the orbit remains the same. 
In a circle the motion is uniform, in an ellipse it varies, but 
in both the planet arrives at the extremities of the major 
axis in the same time. 

§ 471. Since the position of the earth and its observed 
motion have no effect on the time during which a planet ap- 
pears in the ecliptic, the sidereal periods of the planets may 
be obtained with accuracy by thus noting their passages 
through the nodes of their orbit. A very slight retreat 
of the nodes must however be allowed for. The synodical 
revolution brings a planet to the same angular distance 
when viewed from the earth. It differs from the sidereal 
period, which brings the planet to the same situation as re- 
gards the sun. The sidereal is uninfluenced by the motion 
of the earth. 

Mercury's sidereal period is nearly 88 days, his synodical 
is 116 ; that of Yenus is less than 225 days, while her sy- 
nodical revolution occupies about 584 days. The planet 
must not only perform its own sidereal revolution, but it 
must move on to compensate for the arc the earth has 
gone through in the same time. This arc is described by 
the earth with her velocity in the same time that the planet 
with its velocity describes that arc plus 360°. If the 
planet be a superior one we must reverse the calculation. 
The earth will describe 360-j-a, certain arc, while the 
planet describes a similar arc. Let V and v be the 
mean angular velocity, x the arc of excess ; then V : 
v : : 1+x : x ; and V — v : v : : 1 : x, whence x is found, 



334 ELEMENTS OF ASTRONOMY. 

and — = the time of describing x, or the difference be- 

v ° ' 

tween the sidereal and synodical periods. 

§ 472. To ascertain the- relative masses of two bodies, 
we need only compare the force they exercise on bodies re- 
volving; round them. We know that the moon descends 
through 16 T V feet in a minute of time, owing to the earth's 
attraction. We know that the earth descends through 2.2 
this space in a minute of time owing to the sun's attraction. 
But we know the distance of the moon from the earth, 
and of the earth from the sun. The latter distance is four 
hundred times the former. 

The whole attraction is in a ratio made up of the ratios 
of the masses directly, and of the squares of the distances 
inversely. Thus putting F for the whole attraction of the 
sun, and f for the whole attraction of the earth, D and d 
for the distances from the sun and from the earth, we 
have 

f : F : : m : M : : D 2 : d 2 
f : F : : md 2 : MD 2 
fd 2 : FD 2 : : m : M. 
m : M : : 1 X l 2 : 2.2 x 400 2 : : 1 : 352000. 

By marking the deflection of one of the satellites of those 
planets which are provided with them, and comparing it 
with the deflection of the moon, the comparative masses of 
the earth and that planet may be found. The masses of 
those planets which have no satellites are proved from their 
perturbations. We can observe how much it influences 
another known body at a given distance, and how much it 
is influenced by that body. 

§ 473. The earth's real diameter and a planet's appa- 
rent diameter and distance being known, the planet's real 
diameter and solid contents or size can be calculated. Its 
size and mass being known, its density may be compared 
with that of other bodies. 

If we know the density of one planet we may find that 
of another. Suppose two planets, A and B are found to 



ELEMENTS OF ASTRONOMY. 335 

operate on a third by attractions proportionate to the num- 
bers 7 and 2, at distances which are to each other as 4 to 
3 ; let, moreover, their diameters be as 3 to 2. 

Let d stand for the density of A, and D for that of 
B ; then the attraction of A and B on the third body will 
be directly as their masses, and inversely as their distances 
squared. Hence, remembering that similar solids are to 
each other as the cubes of their diameters, if homogeneous, 
or in the compound ratio of their cubes and their densities, if 

their densities differ, the attraction of A will — X 

and that of B =?!><?. 16 

9 

But by observation the attractions are found to be as 
7 to 2 : therefore, 

3 3 xdx9 : 2 3 xDxl6 : : 7 : 2; or 
486 d = 896 D, that is, 
d : D : : 896 : 486. 



CHAPTER XXI. 

THE PLANETS — (CONTINUED.) 

Mercury. Its rare visibility. Its Phases. Venus. Its brilliancy. Rota- 
tion and Atmosphere. Mars. Its Polar Spots. The Asteroids. Con- 
jecture as to their Origin. Jupiter. Its Belts and Rotation. The Sat- 
ellites of Jupiter. Theory of their Motion. Saturn Its Rings and 
Satellites. Uranus. Neptune. 

Mercury. 

§ 474. Mercury is the nearest planet to the sun ; it- 
appears after sunset in the west with a small but very 
brilliant disc. As long as it is east of the sun it remains 
visible after him in the evening. As soon as in its orbit 
it becomes west of the sun, it appears before the sun in the 
morning, withdraws more and more from him till it is 28° 
distant, then draws nearer daily, till at last it disappears, 



336 ELEMENTS OF ASTRONOMY. 

to become again an evening star. The brief duration of 
its appearance each night is owing to its closeness to the 
sun, from which it is only 37,000,000 miles distant. It 
is .06 the size of the earth, has a day of 24 hours, and a 
year of 87 days, and travels 111,000 miles an hour. Its 
orbit, which is always included within that of the earth, is 
very much inclined to the plane of its equator, and makes 
with the plane of the ecliptic an angle of 7°. 

§ 475. Sometimes when Mercury plunges into the 
sun's rays, he is seen like a black spot traversing the sun's 
disc ; this is called the transit or passage of Mercury. 
His dark appearance during these transits proves that he 
borrows his light from the sun. His transits make it easy 
to determine his inclination to the ecliptic, and consequently 
his nodes. The line he describes in passing over the sun 
is sufficient to determine these. His transits always take 
place either in May or November, but much more fre- 
quently in the latter month, because his orbit being quite 
eccentric he is much nearer the sun in our winter than in 
our spring. The sun forms the base of a cone of luminous 
rays passing to the eye of the observer, and the nearer the 
base Mercury passes the more probability is there of his 
intercepting the rays. 

Mercury is of a spherical form, and exhibits phases like 
the moon. One horn of its crescent has been seen in- 
dented, and this proves the irregularity of its sur- 
face, and likewise gives the time of its rotation. It is 
caused by some mountain on its surface, which in some 
positions screens from our view some of the points illumined 
by the sun. 

No spots have been seen on his surface, and nothing is 
known of his atmosphere, except that it contains clouds. 
It has a pale silvery light, and a slight bluish tint. It 
probably received its name from its swiftness, in which it 
resembles the messenger of the gods. For its motion 
from point to point of space is more rapid than that of any- 
other planet. 

§ 476. The sun transmits to it seven times the heat 
of our torrid zone. The mean heat must be above that of 
boiling quicksilver, and water would boil even at his poles 



ELEMENTS Otf ASTRONOMY. 337 

unless the sun's rays are modified by an atmosphere pecu- 
liar to the planet. 

When Mercury and Yenus are nearest the earth, or in 
their inferior conjunction, their dark side is toward it, and 
unless they pass over the sun's disc they are invisible. 
As they revolve in different planes from the earth, transits 
can only take place when they are in or near their nodes 
at the time of their inferior conjunction. 

As they move forward in their orbits they gradually 
show a portion of their enlightened surface till they reach 
their superior conjunction, when the whole of the bright 
side is turned toward the earth. After which the illumi- 
nated surface decreases in size as they return toward the 
earth. The period of greatest brilliancy of course cannot 
be when they are nearest the earth, neither is it when they 
are most distant ; but it is at a point of their orbits in 
which the smaller extent of the illuminated surface is more 
than compensated by the position of the planet nearer to 
the earth. 

Venus. 

§ 477. Venus is the most beautiful of the planets,, 
and for this reason received the name she bears. She is a 
morning star from her inferior to her superior conjunction, 
and an evening star from her superior to her inferior con- 
junction. 

She is never more than 48° distant from the sun. She 
appears much larger at one time than at another, because 
her distance from the earth varies so much, and is most 
brilliant when in her quarters once in eighteen months, 
being then seen by day-light, though not very dis- 
tinctly. She returns to her most brilliant position once in 
eight years, owing to the ratio of her periodic time to that 
of the earth. She is only visible for three or four hours 
in the morning or in the evening, according as she is west 
or east of the sun, being alternately before or after him 
for about 290 days, and appearing sooner than she other- 
wise would owing to her angular velocity being greater 
than that of the earth. 
29 



338 ELEMENTS OP ASTRONOMY. 

When a morning star, she rises before the sun, and is seen 
from the earth in the form of a handsome crescent, with its 
convex side turned eastward toward him. When an eve- 
ning star, she follows the sun, and becomes visible after he 
has set, having the convex side of her crescent turned 
toward him. 

Mercury is her morning and evening star, as she is 
ours. 

§ 478. The mean distance of Venus from the sun is 
70,000,000 miles. Her size compared to that of the earth 
is t 9 o 3 q. Her day is 23 hours long, and her year includes 
224 of our days, or 234 of her own. Her orbit is inclined 
3° to the the ecliptic, and her axis of rotation is inclined 
to her orbit 75°, that is 51° more than the earth's axis is 
inclined to the ecliptic. The greatest declination of the 
sun on each side of her equator is 75°. Her tropics are 
15° from her poles, and her polar circles at the same dis- 
tance from her equator. She has therefore at her equator 
two severe winters and two summers in a year. The north 
pole of her axis is inclined toward Aquarius, the earth's 
toward Cancer, consequently the northern hemisphere of 
Venus has summer in the sign in which our earth has win- 
ter, and vice versa. 

The period of the rotation of Venus is determined like 
that of Mercury, by the interval of time between two suc- 
cessive appearances of the truncated horns of her crescent. 
Not only is her crescent truncated at each end, but a little 
bright peak is visible at some distance from the illuminated 
surface. 

§ 479. Venus has a larger atmosphere and more 
marked clouds than Mercury. The existence of the atmo- 
sphere is known not only by the clouds but by its refrac- 
tion. It causes the illuminated portion of her surface to 
look larger than it otherwise would. This dense atmo- 
sphere may perhaps cause the white and silvery appearance 
and the brightness by which Venus is distinguished from 
all other planets. 

The existence of a satellite to Venus is a question which 
has been very much discussed. We are so unfavorably 
placed for seeing a satellite of hers that she may have one 



ELEMENTS OF ASTRONOMY. 339 

undiscovered by us. Its enlightened side can never be 
fully turned toward us but when Venus is in her superior 
conjunction 160,000,000 miles distant from us, and Venus 
then appears but little larger than an ordinary star. When 
she is between us and the sun, her moon would have its 
dark side toward us, and we could not see it any more than 
our own moon at time of change. It is however improba- 
ble that Venus has a moon, as modern instruments would 
have revealed one not extremely minute. 

Mars. 

§ 480. This planet lies next outside of our earth. Its 
mean distance from the sun is 146,000,000 of miles. Its 
distance from the earth is so variable that its diameter ap- 
pears at some times five times as large as at others. It 
moves at the rate of 55,000 miles an hour, and performs 
its annual circuit in an eccentric ellipse in 686 of our days. 
By the spots on its disc it is found that it rotates on its 
axis in 24 hours. Its axis of rotation is inclined to that of 
its orbit 30°, and the inclination of its orbit to the ecliptic 
is 1° 51'. Its equatorial is to its polar diameter as 16 to 
15. Its diameter is about 4,100 miles. 

Mars is known in the heavens by its red dusky appear- 
ance. It has a very thin atmosphere j which allows the red 
body of the planet to be perceived with its unvarying 
marks. The outlines of what may be land and sea are dis- 
tinctly visible, the continents being of a ruddy color, proba- 
bly belonging to the soil, and the seas consequently ap- 
pearing greenish. These spots are not always to be seen, 
owing perhaps to clouds, but when visible they are of the 
same figure and position. 

§ 481. The polar regions of Mars shine with a bril- 
liancy so superior to the rest of his disc that they appear 
like segments from a larger globe. As each pole appears 
after a long winter this lustre is observed, and is attributed 
to reflection from the snow and ice collected during a win- 
ter night of twelve months' duration. The brightness grad- 
ually disappears after exposure to the sun, and the pole 



840 ELEMENTS OF A&TRONOMY. 

which has been twelve months under the sun's rays is 
nowise distinguishable from the rest of the planet. 

Mars is the only superior planet which exhibits any per- 
ceptible phase ; it never appears as a crescent, but has 
sometimes a moderately gibbous appearance, the enlighten- 
ed portion of the disc being never less than seven-eighths 
of the whole. By means of this gibbosity the proportional 
distances of Mars from the sun, and the earth from the 
sun, may be ascertained. Let E (Fig. 13, Plate II.) be 
the earth at its apparent greatest elongation from the sun 
S, as seen from Mars M. In this position the angle S M E 
is at its maximum. S M bisects the illuminated hemisphere, 
E M bisects the gibbous part ; the diameter of the illumi- 
minated surface gives us S M E. This angle being found, 
the proportion of S E to S M may be found, and it appears 
that S E (the distance of the earth) is § of S M (the dis- 
tance of Mars from the sun) . 

The Asteroids. 

§ 482. The regularity of the intervals between the 
other planets, and the distance between Mars and Jupiter, 
first made it suspected that there might be another planet 
between these. Four of them, Ceres, Pallas, Juno, and 
Vesta, were discovered near the beginning of the present 
century, and since 1845, five other planets, Astraea, Iris, 
Hebe, Flora and Metis, -have been added to the group. 
These are sometimes called the ultra zodiacal planets, be- 
cause their orbits are so much inclined to the ecliptic as 
to pass beyond the zodiac. This circumstance and their 
small size have given rise to the conjecture that they may 
be the fragments of some larger planet which once existed 
at this distance from the sun. It seems more probable 
that they took their present form at the same time with the 
other members of the solar system. They seem a con- 
necting link between the planets and the zone of meteoric 
bodies which we have before described. They may thus 
be regarded as the smallest of the planets, or the largest 
of the planetoids. 

Owing to their small size and their distance from us 



ELEMENTS OE ASTRONOMY. 341 

very little is known of them. They are never visible to 
the naked eye, and through the telescope appear like small 
nebulous stars. Some of them show a disc so that their 
size is known, others have as yet shewn no disc. Their 
diameters vary in size from -3V to T <^ that of the earth, 
or even less than this. Their orbits vary in inclination to 
one another as much as 8°. 

§ 483. Ceres was the first discovered. It revolves in 
4 J years, in an orbit inclined 10° to the ecliptic. It is 
264 millions of miles from the sun, and appears like a 
nebulous star surrounded by very variable mists. 

The orbit of Pallas is extremely elongated, and its incli- 
nation to the ecliptic is 34°, greater than that of any other 
planet. Its distance is 267,000,000 miles. It is of a 
whitish color and nebulous appearance, indicating an ex- 
tensive and vaporous atmosphere, little repressed and con- 
densed by the attraction of so small a mass. Its diameter 
does not much exceed 79 miles, so that a steam-carriage 
might go round it in a few hours. 

Juno has a year equal to 4J of ours, and a distance of 
256,000,000 miles. Its orbit is inclined 23°. 

Vesta is smaller than those previously discovered. It 
has about the same surface as the kingdom of Spain. It is 
only 225,000,000 miles from the sun ; its orbit is inclined 
7° ; its year is 3 J. 

Astraea differs but little from Juno in size, mean dis- 
tance, and periodic time. It has as yet shown no disc. 

Flora equals a star of the eighth or ninth magnitude, 
and shines with a bluish light. 

Jupiter. 

§ 484. Jupiter is the largest of the planets, and next 
to Yenus the most brilliant. Jupiter's disc is crossed in 
one direction by dark bands or belts, which vary in breadth 
and situation. These bands have the appearance of strings 
of clouds. Between the bands, in the dark spaces, bright 
spots, on the surface of the planet, are visible. These 
clouds probably owe their form to currents analagous to 
our trade winds. Such currents must be much stronger 
in Jupiter than in the earth, owing to Jupiter's greater 
29* 



342 ELEMENTS OF ASTKONOMY. 

rapidity of rotation, particularly about the equator. Jupi- 
ter has probably very little change of seasons for his orbit 
is very slightly inclined to the plane of his rotation. His 
day, we have already said, is very short, only 10 hours in 
length, while his year contains 10,000 of his days, and 
4,332 of ours. His diameter is 89,000 miles. 

§ 4S5. * " The changes which take place in the planet- 
ary system are exhibited on a smaller scale by Jupiter and 
his satellites ; and, as the period requisite for the develope- 
naent of the inequalities of these moons only extends to a 
few centuries, it may be regarded as an epitome of that 
grand cycle which will not be accomplished by the planets 
in myriads of ages. The revolutions of the satellites about 
Jupiter are precisely similar to those of the planets about 
the sun : it is true they are disturbed by the sun, but his 
distance is so great that their motions are nearly the same 
as if they were not under his influence. The satellites, 
like the planets, were probably projected in elliptical orbits ; 
but, as the masses of the satellites are nearly one hundred 
thousand times less than that of Jupiter ; and as the com- 
pression of Jupiter's spheroid is so great, in consequence 
of his rapid rotation, that his equatorial diameter exceeds 
his polar diameter by no less than six thousand miles ; the 
immense quantity of prominent matter at his equator must 
soon have given the circular form observed in the orbits of 
the first and second satellites, which his superior attraction 
will always maintain. The third and fourth satellites being 
farther removed from his influence, revolve in orbits with a 
very small eccentricity. And although the first two sensi- 
bly move in circles, their orbits acquire a small ellipticity, 
from the disturbances they experience. 

§ 486. It has been stated, that the attraction of a 
sphere on an exterior body is the same as if its mass were 
united in one particle in its centre of gravity, and there- 
fore inversely as the square of the distance. In a sphe- 
roid, however, there is an additional force arising from the 
bulging mass at its equator, which acts as a disturbing 
force. One effect of this disturbing force in the spheroid 

*From Mrs. Somerville's Connection of the Physical Sciences. 



ELEMENTS OP ASTKONOMY. 343 

of Jupiter is, to occasion a direct motion in the greater axes 
of the orbits of all his satellites, which is more rapid the 
nearer the satellite is to the planet, and very much greater 
than that part of their motion which arises from the dis- 
turbing action of the sun. The same cause occasions the 
orbits of the satellites to remain nearly in the plane of 
Jupiter's equator, on account of which the satellites are 
always seen nearly in the same line ; and the powerful 
action of that quantity of prominent matter, is the reason 
why the motions of the nodes of these small bodies is so 
much more rapid than those of the planet. 

§ 487. The nodes of the fourth satellite accomplish a 
tropical revolution in 531 years ; while those of Jupiter's 
orbit require no less than 36,261 years ; a proof of the re- 
ciprocal attraction between each particle of Jupiter's equa- 
tor and of the satellites. In fact, if the satellites moved 
exactly in the plane of Jupiter's equator, they would not 
be pulled out of that plane, because his attraction would be 
equal on both sides of it. But, as their orbits have a 
small inclination to the plane of the planet's equator, there 
is a want of symmetry, and the action of the protuberant 
matter tends to make the nodes regress by pulling the 
satellites above or below the planes of their orbits ; an 
action which is so great on the interior satellites, that the 
motions of their nodes are nearly the same as if no other 
disturbing force existed. 

The orbits of the satellites do not retain a permanent in- 
clination, either to the plane of Jupiter's equator, or to 
that of his orbit, but to certain planes passing between the 
two, and through their intersection. These have a greater 
inclination to his equator the farther the satellite is re- 
moved, owing to the influence of Jupiter's compression ; 
and they have a slow motion corresponding to secular varia- 
tions in the planes of Jupiter's orbit and equator. 

§ 488. The satellites are not only subject to periodic 
and secular inequalities from their mutual attraction, simi- 
lar to those which affect the motions and orbits of the 
planets, but also to others peculiar to themselves. Of the 
periodic inequalities, arising from their mutual attraction, 
the most remarkable take place in the angular motions of 



344 ELEMENTS OE ASTRONOMY. 

the three nearest to Jupiter, the second of which receives 
from the first a perturbation similar to that which it pro* 
duces in the third ; arid it experiences from the third a 
perturbation similar to that which it communicates to the 
first. In the eclipses these two inequalities are combined 
into one, whose period is 437 — 659 days. The variations 
peculiar to the satellites, arise from the secular inequalities 
occasioned by the action of the planets in the form and po- 
sition of Jupiter's orbit, and from the displacement of his 
equator. 

§ 489. It is obvious that whatever alters the relative po- 
sitions of the sun, Jupiter, and his satellites, must occasion a 
change in the directions and intensities of the forces, which 
will affect the motions and orbits of the satellites. For this 
reason the secular variations in the eccentricity of Jupiter's 
orbit occasion secular inequalities in the mean motions of 
the satellites, and in the motions of the nodes and apsides 
of their orbits. The displacement of the orbit of Jupiter, 
and the variation in the position of his equator, also affect 
these small bodies. The plane of Jupiter's equator is in- 
clined to the plane of his orbit at an angle of 3° 5' 30", 
so that the action of the sun and of the satellites them- 
selves produces a nutation and precession in his equator 
precisely similar to that which takes place in the rotation 
of the earth, from the action of the sun and moon. Hence 
the protuberant matter at Jupiter's equator is continually 
changing its position with regard to the satellites, and pro» 
ducing corresponding nutations in their motions. And, as 
the cause must be proportional to the effect, these inequali- 
ties afford the means, not only of ascertaining the compres* 
sion of Jupiter's spheroid, but they prove that his mass is 
not homogeneous. Although the apparent diameters of the 
satellites are too small to be measured, yet their perturba- 
tions give the values of their masses with considerable ac- 
curacy—a striking proof of the power of analysis. 

§ 490. A singular law obtains among the mean mo- 
tions and mean longitudes of three satellites. It appears 
from observation that the mean motion of the first satellite, 
plus twice that of the third, is equal to three times that 
of the second ; and that the mean longitude of the first 



ELEMENTS OE ASTRONOMY. 845 

satellite, minus three times that of the second, plus twice 
that of the third, is always equal to two right angles. It 
is proved by theory, that if these relations had only been 
approximate when the satellites were first launched into 
space, their mutual attractions would have established and 
maintained them, notwithstanding the secular inequalities 
to which they are liable. They extend to the synodic 
motions of the satellites ; consequently they affect their 
eclipses, and have a very great influence on their whole 
theory. The satellites move so nearly in the plane of Ju- 
piter's equator, which has a very small inclination to his 
orbit, that the first three are eclipsed at each revolution by 
the shadow of the planet : the fourth satellite is not eclipsed 
so frequently as the others. The eclipses take place <close 
to the disc of Jupiter when he is near opposition ; but at 
times his shadow is so projected with regard to the earth, 
that the third and fourth satellites vanish and reappear on 
the same side of the disc. These eclipses are in all re- 
spects similar to those of the moon ; but, occasionally, the 
satellites eclipse Jupiter, sometimes passing like obscure 
spots across his surface, resembling annular eclipses of the 
sun, and sometimes like a bright spot traversing one of his 
dark belts. Before opposition, the shadow of the satellite, 
like a round black spot, precedes its passage over the disc 
of the planet, and after opposition, the shadow follows the 
satellite. 

§ 491. In consequence of the relations already men- 
tioned in the mean motions and mean longitudes of the first 
three satellites, they never can be all eclipsed at the same 
time. For when the second and third are in one direc- 
tion, the first is in the opposite direction ; consequently, 
when the first is eclipsed, the other tw T o must be between 
the sun and Jupiter. The eclipses of Jupiter's satellites 
have been the means of a discovery which, though not 
immediately applicable to the wants of man, unfolds one of 
the properties of light — that medium without whose cheer- 
ing influence all the beauties of creation would have been 
to us a blank. It is observed, that those eclipses of the 
first satellite, which happen when Jupiter is near conjunc- 
tion, are later by 16' 26". 6 than those which take place 



346 ELEMENTS OF ASTRONOMY. 

when the planet is in opposition. As Jupiter is nearer to 
us when in opposition by the whole breadth of the earth's 
orbit than when in conjunction, this circumstance is attri- 
buted to the time employed by the rays of light in crossing 
the earth's orbit, a distance of about 190,000,000 miles ; 
whence it is estimated that light travels at the rate of 
190,000 miles in one second. Such is its velocity, that 
the earth, moving at the rate of nineteen miles in a second, 
would take two months to pass through a distance which a 
ray of light would dart over in eight minutes. 

§ 492. The velocity of light deduced from the ob- 
served aberration of the fixed stars, perfectly corresponds 
with that given by the eclipses of the first satellite. The 
same result, obtained from sources so different, leaves not a 
doubt of the truth. Many such beautiful coincidences, 
derived from circumstances apparently the most unpromis- 
ing and dissimilar, occur in physical astronomy, and prove 
connections, which we might otherwise be unable to trace. 
The identity of the velocity of light, at the distance of Ju- 
piter, and on the earth's surface, shows that its velocity is 
uniform ; and if light consists in the vibrations of an elastic 
fluid or ether filling space, an hypothesis which accords 
best with observed phenomena, the uniformity of its velocity 
shows that the density of the fluid throughout the whole 
extent of the solar system must be proportional to its elas- 
ticity." 

Saturn, 

§ 493. A still more wonderful mechanism is displayed 
in Saturn, the planet next in order of remoteness to Jupi- 
ter, and not much inferior to him in magnitude, being 
79,000 miles in diameter, and 1,000 times the bulk of the 
earth. Seen by the naked eye, it is a star of a dull lus- 
tre, and as its motion is very slow, scarcely distinguishable 
from a fixed star. At a distance of 915,000,000 of miles 
from the sun, it moves in an orbit inclined only 2° to the 
ecliptic. Its revolution occupies twenty-four years, its ro- 
tation about ten hours. The flattening of its poles amounts 
to one twelfth of its diameter. It is surrounded by a ring 



ELEMENTS OF ASTRONOMY. 347 

which revolves nearly in the plane of its equator. This 
ring is about 60,000 miles broad, and 100 thick, and is 
about 19,000 miles distant from the planet. When this 
ring was first seen only the ends of it were visible, looking 
like the ears of a jar. It was said that Saturn was old 
and had two helpers to uphold him. As the planet passed 
on in its orbit, the ring widened out, and showed itself cir- 
cular. It was then thought to be a necklace of moons, or 
two moons connected by a line of light. Afterward a dark 
stripe, concentric with the circumference of the ring, was 
distinctly visible, and was thought to be a deep valley. 
Herschel saw in this dark interval a shining spot, unlike a 
mountain, and at last detected it to be a star, and knew 
that the ring was composed of two parts. 

§ 494. It has been thought that it consists of many 
concentric rings, which revolve with a rapidity sufficient to 
counteract the attraction of Saturn. They rotate from 
west to east, and in the same time a satellite at an equal 
distance would require. That the rings are solid and 
opaque is shown by their casting their shadow on the body 
of the planet when they are toward the sun, and by their 
receiving the shadow of the planet on that part of them 
which is remote from the sun. 

Since the rings always remain at right angles to the 
axis of rotation, and that is inclined to the ecliptic 28°, the 
rings must present to an observer on the earth a great 
variety of appearances. They must vary from a long el- 
lipse to a mere line. When the plane of the rings passes 
through the sun's centre, only the edge of the rings is 
illuminated ; when their plane passes through the earth's 
centre, they appear as a very fine line drawn across the 
disc and projecting out on each side. The rings disappear 
to common telescopes once in fifteen years, but the disap- 
pearance is generally double, the earth passing twice through 
the plane of the rings before they are carried past our orbit 
by the slow motion of Saturn. Then the two. inner satellites 
appear like beads threading the slender line of light to 
which the rings are reduced. They separate from them at 
each end while turning round on their orbits. They move 
close to the edge of the rings, and their orbits never deviate 



348 ELEMENTS OE ASTRONOMY. 

much from their plane. To those regions of the planet which 
lie above the enlightened side of the rings they must ap- 
pear like wide arches spanning the heavens from horizon to 
horizon, while the regions beneath their shade suffer an 
eclipse of fifteen years. Each side of these rings has fif- 
teen years of light and fifteen of darkness. The mass of 
the rings is thought to be T !s- part of that of the planet. 
It is thought that they are not perfectly concentric with 
the planet, but, owing to some inequality in their thickness 
or their density, revolve round a centre of gravity not the 
same as that of the planet. If they were concentric with 
the body of Saturn, any disturbing cause would increase 
their oscillations till their equilibrium would be overthrown ; 
as it is, the heavier parts have sufficient momentum to 
overcome minute disturbing forces — such as the attraction 
of satellites. 

§ 495. ' Saturn has eight satellites. The most distant 
but one is the largest, and is nearly of the size of Mars. 
The orbit of the outer satellite is inclined 80° to the plane 
of the ring, while the orbits of the others nearly coin- 
cide with it, and are by the attraction of the equatorial 
parts made circular. The outer one exhibits periodical 
changes, which prove that like our moon it rotates and 
revolves in the same time. The two interior satellites, 
and that recently discovered, are very minute, and can 
scarcely be seen except when the ring disappears. Their 
periods of revolution vary from twenty- two hours to seventy- 
nine days. 

If the inhabitants of Saturn and Jupiter have such eyes 
as ours, unassisted by instruments, Jupiter is the only 
planet which can be seen from Saturn, and Saturn the 
only one which can be seen from Jupiter. So that the 
inhabitants of these planets must have much better sight 
than we, or have equally good instruments, to find out that 
there is such a body as the earth in the universe. Tor 
the earth is but little larger, seen from Jupiter, than his 
outer moon is seen from the earth ; and if his large body 
had not first attracted our sight and caused us to view him 
through a telescope, we should probably never have known 
the existence of his moons or those of Saturn. 



ELEMENTS OF ASTRONOMY. 349 



Uranus. 



§ 496. Sir William Herschel, looking through his 
seven-feet telescope at the stars in the feet of Gemini, saw 
a little star rather different from others of the same light, 
and apparently larger, which he suspected of being a 
comet. He looked at it with a power of 932, and found 
its diameter still increasing. He compared it with many 
little stars, and observed its position with regard to them. 
Two days after he was assured it was no star, because 
it had changed its place. By degrees it was found 
that its diameter did not change sensibly, that its orbit 
was almost round. After a year's observation it was 
found that its revolution occupied about eighty-three years* 
It was recognized as a star which in different catalogues 
had been set down successively in two different places. 
Its distance from the sun is 1,840,000,000 of miles. Its 
orbit almost coincides with the ecliptic. Its time of rota- 
tion is not known ; nor its compression ; but the orbits of 
its satellites are nearly perpendicular to the plane of the 
ecliptic, and by analogy they should be in the plane of its 
equator. Of the physical constitution of Uranus nothing 
is known. The earth cannot be visible, even as a tele- 
scopic body, to an object so remote. It can be seen from 
the earth only by excellent eyes, and its satellites only by 
an instrument far better than is commonly met with in 
observatories ; their number is uncertain. 

Neptune. 

§ 497. Of this planet, owing to its remoteness and re- 
cent discovery, little is known. It shines with a bluish 
light, and is accompanied by one and perhaps more satel- 
lites. Its mass is about ^ <bo of that of the sun. The 
extraordinary circumstance of its having been discovered 
in consequence of the predictions of two eminent astrono- 
mers, gives to this planet a very prominent place in astro- 
nomical history. 

30 



350 ELEMENTS OF ASTRONOMY. 



CHAPTER XXII. 

THE MOON. 

Size and Mass of the Moon. Its Distance and Period. Revolution of the 
Nodes of the Lunar Orbit. Appearance of the Moon. Libration. 
Phases of the Moon. The Harvest Moon. The Lunar Theory. Action 
of the Sun. Evection. Variation. Annual Equation. Action of the 
Planets. Acceleration of the Mean Motion. 

§ 498. The mass of the moon is determined from 
several sources ; from her action on the terrestrial equator, 
which causes nutation in the axis of rotation ; from an ine- 
quality she produces in the sun's longitude ; and from her 
action on the tides : it appears to be about one eightieth part 
of that of the earth. Since her volume is nearly one fiftieth 
that of the earth, her density is two thirds that of the earth. 
Her form is slightly spheroidal. Her diameter is 2,160 
miles, and her average distance from the centre of the 
earth but 237,360 miles. She completes the circuit of the 
heavens in 27 days, 7 hours, 43 minutes, moving at the 
rate of 2,000 miles an hour, in an orbit whose eccentricity 
is about 12,985 miles. By observation of her parallax it 
is found that her mean distance is about sixty times the 
radius of the earth. Her greatest distance is 64, her 
least 56 radii of the earth, quantities which are to each 
other as 8 to 7, and which give a much greater eccentricity 
than that of the solar ellipse. Her greatest apparent diam- 
eter is 33' 31", her least 29' 22". Beside the variation in 
her diameter, owing to the ellipticity of her orbit, there is 
a slighter one owing to parallax. When the moon is in the 
zenith, she is nearer to an observer by the radius of the 
earth, or one sixtieth of her whole distance, than when in 
the horizon. Her diameter is accordingly 30" larger than 
when in the horizon. Her orbit is inclined to the ecliptic 
a little more than 5°- ; this inclination varies ; but it never 
falls short of 5°, nor exceeds 5° 18'. When she crosses 
north of the ecliptic she is in her ascending node, when she 
passes south of it she is in her descending node. Her 



ELEMENTS OF ASTRONOMY. 351 

nodes retreat on the ecliptic 19° 2V a year, and complete 
the circuit of the heavens in 6,793d.' lOh. 6' 30". The 
points in the orbit in which the moon is nearest to and far- 
thest from the earth, are called respectively its apogee 
and perigee ; the line joining them is called the line of ap- 
sides. This line advances at the mean rate of 40° 40' 32" 
every year, and completes the circuit of the heavens in 
3,232d. 13h. 56' 17". 

The periodic time of the moon, or that occupied in 
making a complete revolution round the earth, or to the 
same star, occupies 27d. 7h. 43' 12". Her synodic period 
or time of being again in the same direction with the sun, 
occupies 29d. 12h. 46' 3". 

§ 499. Either of these being known, the other may 
be computed from it. Thus let s represent the synodic 
period of the moon, p her periodic time, P the period of a 
revolution of the sun ; and let A represent the angle 
through which the sun has moved before the moon over- 
takes him ; and also let us suppose for the present that the 
angular motions both of the sun and moon are uniform. 
If this be the case, as the sun moves through 360° in P, 

s 
he will move through 360° -=- in the time s, or in the syno- 

. s 

die period of the moon ; or the angle A will be 360°— 

But the moon, before she overtakes the sun, will have 
moved through 360° +a, or 360° +360° -J ; and as she 
takes the time p to move through 360°, she will take the 
time — - t ~t" p to move through 360°+ A: 



360° 



or S: 



5^!_+?^°_i p p + and hence p== Zi 

360° ^ ■ ^ ^ P-f-s 

or s= p— — ; equations from which, if either quantity s 

or p be known, the other may be computed ; for P, the 
length of the year is known. The angular motions both of 
the sun and moon being variable, this is not an accurate 
method of estimating these elements. It gives however 
their mean value. 



352 ELEMENTS OF ASTRONOMY. 

The synodic period may be best learned from eclipses of 
the moon. The middle of such an eclipse is very near the 
time in which the earth is directly between the sun and 
moon, and that exact time may be computed from observa- 
tions made of the eclipse. From one eclipse to another, 
happening under similar circumstances, must be an exact 
number of synodic periods of the moon. The number of 
such periods which have elapsed is known, and the whole 
interval between the two eclipses, divided by this, gives the 
exact length of the mean synodical period. The position 
of the moon's apogee in the two eclipses, and the season 
of the year in which they are observed, introduces some 
slight incorrectness, but divided among so many months, 
the error is unimportant. From the synodical period thus 
found, subtract its excess over the sidereal period. This 
excess is the arc the sun passes through in 29.53 days, or 
29°. 1. The moon, moving at the rate of 13° 17' a day, 
will describe this arc in 2.21 days. 

The moon's year includes thirteen of her synodic days, 
and each day has a fortnight's light and a fortnight's dark- 
ness. 

§ 500. The moon's disc exhibits to the naked eye 
numerous irregularities ; observed through a telescope it is 
covered with crater-like hollows and with steep jagged hills. 
Its face is as well known to astronomers as the face of the 
earth, and like that is mapped down and designated by 
names. As these features remain unchanged, it is known 
that the moon always has the same hemisphere to us. 
Some of the cavities and basins are extremely bright, and 
have the power of reflecting light more brilliantly than 
others, just as the mountains and deserts of the earth are 
more reflective than the meadows and valleys. A green 
color prevails in many parts of the moon, probably the 
color of the rock. The height of the mountains, and the 
depth of the hollows, are known by the shadows cast when 
the sun is rising or setting on them. Before the sun's rays 
reach the general surface it lights up these peaks precisely 
as on earth. These mountains are less elevated than our 
highest, but are higher in proportion to the moon's size, 
some of them being about 3 or 4 miles high. 



ELEMENTS OP ASTRONOMY. 353 

It has been asserted that works of art have been and 
may be seen in the moon. The improbability of this may 
be seen from these facts. The smallest portion of the 
moon's surface perceptible by the naked eye is equal to 
about seventy lunar miles ; a distance of a mile in the 
moon subtends at the eye only an angle of one second. 

The question of the moon's atmosphere has been much 
discussed, and there is much contradictory evidence as to 
its existence. If there is one it is probably very rare. 
There is good evidence that there is no water in the moon, 
for the edge of her crescent is always more or less jagged, 
whereas the presence of seas or lakes would make it par- 
tially smooth. No clouds have ever been discovered. 

§ 501. Since the axis of the moon is a degree and a 
half inclined to the axis of her revolution, and moreover as 
her revolution takes place in a plane inclined to the ecliptic, 
we sometimes see a little beyond one or the other pole. 
Sometimes, for the same reason, we see less of the illu- 
mined portion than we might expect. When she is full in 
the highest part of her orbit, a deficiency appears in the 
lower part of her disc, because we have not a full view of 
the enlightened hemisphere. When she is full in the 
lowest part of her orbit, there must be a similar deficiency 
observed in the upper edge. This is called the moon's 
lib ration in latitude. 

She has also a libration in longitude, owing to her vary- 
ing speed in her orbit, while her rotation is equable. In 
this way a strip a little west or east of the hemisphere 
usually presented is brought into view. 

There is still another phenomenon of the same kind. 
The part of the moon presented to an observer at any 
place is bounded by a circle perpendicular to the line join- 
ing his place and the centre of the moon. To observers 
at different places, therefore, appearances in some degree 
different will be presented ; for the moon is not so distant 
from the earth but that the lines joining her centre with 
different points on the earth's surface may make a sensible 
angle ; in the extreme case, not less than twice the hori- 
zontal parallax of the moon, or nearly 2° on an average. 
30* 



354 ELEMENTS OF ASTRONOMY. 

Every day presents another parallactic change in the 
moon's appearance, arising not from the different positions 
of the observers, but from the different positions of the 
moon as seen by an observer. When the moon rises in the 
east, an observer sees a little more of her western and then 
upper surface, than he would see if placed at the centre of 
the earth. When she is setting in the west, he sees a little 
more of her eastern and then upper side. This is called 
the diurnal libration. 

Owing to all these causes combined, and also to a slight 
nutation of the moon's axis, we get sight of a zone a few 
degrees in breadth beyond an exact hemisphere of the 
moon. 

§ 502. The sun and stars rise and set to an inhabitant 
of the moon as they do to us, but only once a month in- 
stead of once in twenty-four hours. Since there is no at- 
mosphere in the moon sufficiently dense to reflect light, 
the heavens in the day-time must have the appearance of 
night, and the stars must appear on a black ground, and 
as bright as they do in the night to us. Seen from the 
moon the earth must appear the largest body in the uni- 
verse ; its disc must be thirteen times as large as that of 
the moon seen from the earth. Since the moon's rotation 
would give the earth an apparent motion westward pre- 
cisely equal to that which its revolution would give it east- 
ward, the earth with regard to the moon will appear to 
stand still. It will be always visible in the same part of 
the heavens, though not among the same stars. Its only 
motion will be a slight apparent balancing caused by the 
libration of the moon. 

It will always be invisible to one hemisphere of the moon 
and be continually seen from the other. Those to whom it 
is visible will see it exhibit all the phases which the moon 
presents to us, only at different times. The light received 
from the earth must prevent that hemisphere of the moon 
which is turned toward it from ever being in total dark- 
ness. It has alternately a fortnight of earth-light, and a 
fortnight of sun-light. But the other hemisphere of the 
moon has a fortnight's sun-light and a fortnight's darkness 
alternately. Hence its inhabitants, if there be any, can 



ELEMENTS OF ASTKONOMY. 355 

never see the earth unless they travel to gratify their 
curiosity. 

While the earth turns on its axis the aspect it presents 
to the moon must be very various. Our seas, continents, 
forests and islands, must appear as so many spots of va- 
rious brilliancy, and the atmosphere with its clouds must 
give still greater variety. 

§ 503. The light reflected from the earth to the moon 
must be very considerable since it is quite perceptible when 
again reflected back to us. The appearance called the new 
moon in the old moon's arms is the opaque body of the moon 
made visible by the light sent from the earth. It is seen only 
when the crescent of the moon is small. As the illumined 
portion of the moon seen by us increases it overpowers this 
ashy light, and the earth waning at the same time actually 
sends it less light. It has been thought that the moon 
possesses an innate light, but it is more probable that this 
light is refracted by the earth's atmosphere, and the state 
of the earth's atmosphere causes it to assume different hues 
in different eclipses. In some eclipses the moon retains al- 
most all her light, usually however appearing of a reel cop- 
pery hue. This may be owing to electrified vapors belong- 
ing to the earth's atmosphere, and interposed between it 
and the moon. Instances are recorded however where this 
feeble light has been entirely absorbed, so that the moon 
has altogether disappeared in her eclipses. It is not how- 
ever certain that all the light of the moon when eclipsed 
comes from the earth. 

§ 504. The monthly phases of the moon are caused 
by her changes of position with regard to the earth. One 
half of her surface is always illumined, and as she revolves 
round us we see a larger or less portion of her illuminated 
surface. When her light is greatest she is most distant 
from the sun, and therefore upon the horizon longer than at 
any other time. When her light is least, when she shows but 
a portion of her illuminated surface, she remains visible 
above the horizon but a few hours after sunset or before sun- 
rise. She is of course above the horizon as many hours on the 
average one solar day as another, but when she is full all 
these hours are night hours, when she is waxing or waning 



356 ELEMENTS OP ASTRONOMY. 

a portion of them are hours of daylight, and she is scarcely 
noticed except when pretty large in a pale wintry sky. 
When the moon is on the meridian at midnight her disc is 
entirely luminous, she is round and brilliant, she becomes 
visible when the sun sets, and sets when he rises. In a 
few days the bright part of her disc diminishes in breadth 
on the side farthest from the earth, she rises later and sets 
after sunrise. When she reaches her quarter her disc is 
reduced one half, she appears only the latter half of the 
night. Then she becomes a crescent whose horns are 
turned toward the west away from the sun, she rises later 
and later, the crescent wanes, the moon is dark ; she rises 
with the sun and is seen no longer. She is usually invisi- 
ble for several days, but the duration of her invisibility de- 
pends partly on climate, atmosphere, and power of vision. 

§ 505. An instance is on record of a lady, in the pro- 
verbially dingy atmosphere of England, noticing the old 
moon near her conjunction with the sun in the morning ex- 
hibiting a thread-like crescent, and the day after in the eve- 
ning she observed the crescent turned the opposite way 
and eastward of the sun soon after sunset. Thus the same 
person saw, on the morning of one clay and the evening of 
the next, a waning and a waxing moon. In Smyrna, when 
the atmosphere is exceedingly clear, the whole round dark 
blue disc of the moon is visible at the time of conjunction. 

When the new moon appears east of the sun the horns 
of her crescent are turned from him, she remains above 
the horizon but a few hours. The crescent gradually in- 
creases, she follows the sun at a greater distance, becomes 
visible when he sets, and remains visible longer. She 
comes to her quarter, and still increasing, and remaining 
longer visible, at length comes in a line with the earth and 
sun, and is again a full moon. In whatever part of her 
monthly course the moon may be, and at whatever inclina- 
tion to the ecliptic, if a line be imagined joining her horns, 
and bisected by another line at right angles to this, 
and produced beyond the convex part of the moon, the 
latter line will be in the direction of the sun. 

§ 506. The two points in the orbit corresponding to 
the new and full moon respectively are called by the com- 



ELEMENTS OF ASTRONOMY. 357 

mon name of syzygies ; those which are 90° from the sun 
are called quadratures. The full moon is always in oppo- 
sition to the sun, and consequently the full moons of winter 
are as much elevated above the equator as the sun is sunk 
below it, or as the sun is elevated above it in summer, and 
reciprocally it remains no longer above the horizon in sum- 
mer than the sun does in winter. Thus all latitudes be- 
yond the tropics have full moons of great altitude and 
which remain many hours visible in winter, but in summer 
their moons describe a lower and a shorter course. All 
that was formerly said of the sun's appearance to different 
portions of the earth applies to the moon also. These 
are the two full moons which occur about the tropics, 
all the others have a rising and a setting. During the six 
months' day of the poles, the full moon in the opposite part 
of the ecliptic is always invisible ; during their six months' 
night, the moon is visible from quadrature to quadrature, 
circling round the horizon for a fortnight at a time. 

§ 507. The moon's motion among the stars is so rapid 
as to be apparent in the course of a few hours. In the 
course of twenty-four hours she advances nearly 13°, and 
therefore to an observer on the earth rises later every day 
than the day before. If her revolution and our rotation 
were performed in the plane of the ecliptic, the moon 
would rise about three quarters of an hour later each day 
in the year than on the day preceding. But the moon's 
path is inclined to the equator, which causes unequal por- 
tions of it to rise in equal times, and it describes its course 
with a varying rapidity. These causes, however, make 
but a slight difference in the interval between two successive 
risings of the moon, and accordingly, to places on the 
equator, the moon rises about fifty minutes later each day 
than the preceding day. Within the tropics, there is so 
little variety of seasons that no one time is peculiarly har- 
vest time. In higher latitudes, when the whole harvest of 
the year is gathered at once, the autumnal full moons give 
the husbandman an opportunity to complete his labors. 
The full moon of September rises soon after sunset for 
several evenings together, and is called the harvest moon. 
The full moon of October also rises not long after sunset, 



358 ELEMENTS OP ASTKONOMY. 

and nearly at the same time for some nights ; it is called 
the hunter's moon. At the polar circles the autumnal full 
moon rises at sunset during the second and third quarters. 
At the poles the winter moons shine without setting during 
the same quarters. 

§ 508. It is not very easy to imagine the manner in 
which the moon rises at different angles to the horizon of 
places in high latitudes, hut it may be seen on a globe. 
The moon's motion is so nearly in the ecliptic that we may 
consider her as moving in it. Now the different parts of 
the ecliptic, on account of its obliquity to the earth's axis, 
make very different angles with the horizon as they rise 
and set. Those parts or signs which rise with the smallest 
set with the greatest angles, and vice versa. When 
this angle is least a greater portion of the ecliptic rises 
in equal times, than when the angle is larger ; as may be 
seen by elevating a globe to any considerable latitude, and 
turning it round on its axis. Consequently when the moon 
is in those signs which rise or set with the smallest angles, 
she rises or sets with the least difference of time ; and with 
the greatest difference in those signs which rise or set with 
the greatest angles. 

In northern latitudes, the smallest angle made by the 
ecliptic and horizon is when Aries rises, at which time 
Libra sets ; the greatest when Libra rises, at which time 
Aries sets. Therefore the ecliptic rises fastest about Aries, 
and slowest about Libra. On the parallel of London as 
much of the ecliptic rises about Pisces and Aries in two 
hours as the moon goes through in six days ; and therefore 
while the moon is in these signs, she differs but two hours 
in rising for six days together ; that is, about twenty min- 
utes later every day or night in a mean rate. But in four- 
teen days afterward the moon comes to Virgo and Libra, 
which are the opposite signs to Pisces and Aries ; and then 
she differs almost four times as much in rising ; namely, 
one hour and about fifteen minutes. 

§ 509. All these facts may be seen at once, by ele- 
vating the north pole of a globe to any desired altitude, 
and making chalk marks on the ecliptic at intervals of 13°, 
to represent the moon's mean place from day to day. 



ELEMENTS OP ASTRONOMY. 359 

Then turning the globe westward seven of the marks about 
Pisces and Aries will rise in two hours and a half, mea- 
sured by the motion of the index of the hour circle ; but 
about the opposite signs the index will go over eight hours 
in the time that seven marks will rise. The intermediate 
signs will more or less partake of these differences as they 
are more or less remote from these signs. 

Since the rising of the harvest moon is one of the few 
phenomena which may be better understood from a globe 
than by consideration of real events, I will mention one 
other mode of showing it. Let two celestial meri- 
dians on a celestial globe represent the edge of the illu- 
minated hemisphere, and suppose the observer stationed 
on the brazen meridian ; whenever the illumined edge 
comes under the brazen meridian, the observer will be 90° 
from the moon, and will see her rise. But this edge, as 
the globe turns round, will meet the observer at very vary- 
ing intervals, and the nearer the observer is to the polar 
circle the greater will be the differences between these in- 
tervals. 

§ 510. The moon goes round the ecliptic in 27 clays, 
8 hours ; but not from change to change in less than 29 
days, 12 hours ; so that she is in Pisces and Aries at least 
once in every lunation, and in some lunations twice. For 
while the moon goes round the ecliptic, from any conjunc- 
tion or opposition, the earth goes almost a sign forward ; 
and therefore the sun appears to go as far forward, that is 
27£° ; so that the moon must go 27° more than round, and 
as much farther as the sun advances in that interval, 
which is 2 T ^°, before she can again be in conjunction with 
or opposition to the sun. Hence there can be but one con- 
junction or opposition of the sun and moon in a year in any 
particular part of the ecliptic. In the same way the hour 
and minute hands of a watch are never in conjunction or 
opposition in that part of the dial-plate where they were so 
last before. 

§ 511. As the moon can never be full but when she 
is opposite to the sun, and the sun is never in Virgo and 
Libra but in our autumnal months, it is plain that the moon 
is never full in the opposite signs, Pisces and Aries, but 



860 ELEMENTS OF ASTRONOMY. 

in these two months. And therefore we can only have 
two full moons in the year, which rise so near the time of 
sunset for a week together, as above mentioned. 

Here it will probably be asked, why we never observe 
this remarkable rising of the moon but in harvest, seeing 
she is in Pisces and Aries twelve times in the year besides ; 
and must then rise with as little difference of time as in 
harvest ? The answer is plain : for in winter these signs 
rise at noon ; and being then only a quarter of a circle 
distant from the sun, the moon in them is in her first quar- 
ter : but when the sun is above the horizon, the moon's 
rising is neither perceived nor regarded. In spring these 
signs rise with the sun, because he is then in them ; and 
as the moon changes in them at that time of the year, she 
is quite invisible. In summer they rise about midnight, 
and the sun being then three signs, or a quarter of a circle, 
before them, the moon is in them about her third quarter ; 
when rising so late, and giving but very little light, her 
rising passes unobserved. And in autumn these signs, be- 
ing opposite to the sun, rise when he sets, with the moon 
in opposition, or at the full, which makes her rising very 
conspicuous. 

§ 512. In northern latitudes, the autumnal full moons 
are in Pisces and xlries ; and the vernal full moons in 
Virgo and Libra : in southern latitudes, just the reverse, 
because the seasons are contrary. But Virgo and Libra 
rise at as small angles with the horizon in southern latitudes, 
as Pisces and Aries do in the northern ; and therefore the 
harvest moons are just as regular on one side of the equator 
as on the other. 

The moon's oblique motion with regard to the ecliptic 
causes some difference in the times of her rising and sit- 
ting from what is already mentioned. For when she is 
northward of the ecliptic, she rises sooner and sets later 
than if she moved in the ecliptic ; and when she is south- 
ward of the ecliptic, she rises later and sets sooner. This 
difference is variable, even in the same signs, because the 
nodes shift backward about 19|° in the ecliptic every year ; 
and so go round it contrary to the order of signs in 18 
years, 225 days. 



ELEMENTS OF ASTRONOMY. 361 

As there is a complete revolution of the nodes in 19 
years, there must be a regular period of all the varieties 
which can happen in the rising and setting of the moon 
during that time. But this shifting of the nodes never af- 
fects the moon's rising so much, even in her quickest de- 
scending latitude, as not to allow us still the benefit of her 
rising nearer the time of sunset for a few days together 
about the full in harvest, than when she is full at any other 
time of the year. 

§ 513. Superstition has attributed to the moon great 
influence on chemical processes, — on the growth of seeds 
according as they are sown in the waxing or the waning 
moon, — on the weather, and the health and spirits of man- 
kind. Many of the supposed effects are doubtless imagi- 
nary, some are not yet ascertained. Moon beams contain 
chemical rays and rays of heat ; they have an effect similar 
but weaker than the sun's rays in daguerreotypes, so that 
it has been proposed to make the moon daguerreotype her 
own portrait. Sir John Herschel has often been quoted 
as believing that the moon influences the weather. This is 
all the influence he allows her. He thinks that the moon 
when at the full and a few days after, must be in a small 
degree a source of heat to the earth. But this heat ema- 
nating from a body below the temperature of ignition, will 
never reach the earth's surface, but will be arrested and 
absorbed in the upper strata of the atmosphere, where its 
whole power will be expended in converting visible cloud 
to transparent vapor. The rapid dissipation of clouds in 
moderate weather soon after the appearance of a full or 
nearly full moon, which he had himself observed on so 
many occasions, could, he thought, be explained on no 
other principle. 

*§ 514. Several circumstances occur to render the 
moon's motions the most interesting,, and at the same time 
the most difficult to investigate, of all the bodies of our 
System. In the solar system, planet disturbs planet ; but 
in the lunar theory, the sun is the great disturbing cause ; 

* The rest of this chapter is taken from Mrs. Somerville's Connection of 
the Physical Sciences. 

31 



362 ELEMENTS OF ASTRONOMY. 

his vast distance being compensated by his enormous mag- 
nitude, so that the motions of the moon are more irregular 
than those of the planets ; and, on account of the great 
ellipticity of her orbit, and the size of the sun, the approx- 
imations to her motions are tedious and difficult, beyond 
what those who are unaccustomed to such investigations 
could imagine. The moon is about four hundred times 
nearer to the earth than to the sun. The proximity of the 
moon to the earth keeps them together. For so great is 
the attraction of the sun, that if the moon were farther 
from the earth, she would leave it altogether, and would 
revolve as an independent planet about the sun. 

§ 515. The disturbing action of the sun on the moon 
is equivalent to three forces. The first, acting in the di- 
rection of the line joining the moon and earth, increases 
or diminishes her gravity to the earth. The second, act- 
ing in the direction of a tangent to her orbit, disturbs her 
motion in longitude. And the third, acting perpendicu- 
larly to the plane of her orbit, disturbs her motion in lati- 
tude ; that is, it brings her nearer to, or removes her farther 
from the plane of the ecliptic than she would otherwise be. 
The periodic perturbations in the moon, arising from these 
forces, are perfectly similar to the periodic perturbations of 
the planets. But they are much greater and more nume- 
rous ; because the sun is so large, that many inequalities 
which are quite insensible in the motions of the planets, are 
of great magnitude in those of the moon. 

§ 516. Among the innumerable periodic inequalities 
to which the moon's motion in longitude is liable, the most 
remarkable are, the equation of the centre, which is the 
difference between the moon's mean and true longitude, 
the evection, the variation, and the annual equation. The 
disturbing force which acts in the line joining the moon 
and earth produces the evection : it diminishes the eccen- 
tricity of the lunar orbit in conjunction and opposition, 
thereby making it more circular, and augments it in qua- 
drature, which consequently renders it more elliptical. 
The period of this inequality is less than thirty-two days. 
Were the increase and diminution always the same, the 
evection would only depend upon the distance of the moon 



ELEMENTS OF ASTRONOMY. 363 

from the sun ; but its absolute value also varies with her 
distance from the perigee of her orbit. 

§ 517. Ancient astronomers, who observed the moon 
solely with a view to the prediction of eclipses, which can 
only happen in conjunction and opposition, where the ex- 
centricity is diminished by the evection, assigned too small 
a value to the ellipticity of her orbit. The evection was 
discovered about A. D. 140. The variation produced by 
the tangential disturbing force, which is at its maximum 
when the moon is 45° distant from the sun, vanishes when 
that distance amounts to a quadrant, and also when the 
moon is in conjunction and opposition ; consequently that 
inequality never could have been discovered from the 
eclipses ; its period is half a lunar month. The annual 
equation depends upon the sun's distance from the earth ; 
it arises from the moon's motion being accelerated, when 
that of the earth is retarded, and vice versa; for when 
the earth is in its perihelion, the lunar orbit is enlarged by 
the action of the sun ; therefore the moon requires more 
time to perform her revolution. But as the earth ap- 
proaches its aphelion, the moon's orbit contracts, and less 
time is necessary to accomplish her motion — its period, 
consequently, depends upon the time of the year. In the 
eclipses the annual equation combines with the equation of 
the centre of the terrestrial orbit, so that ancient astrono- 
mers imagined the earth's orbit to have a greater eccen- 
tricity than modern astronomers assign to it. 

§ 518. The planets disturb the motions of the moon 
both directly and indirectly ; their action on the earth al- 
ters its relative position with regard to the sun and moon, 
and occasions inequalities in the moon's motion, which are 
more considerable than those arising from their direct ac- 
tion ; for the same reason the moon, by disturbing the 
earth, indirectly disturbs her own motion. Neither the 
eccentricity of the lunar orbit, nor its mean inclination to 
the plane of the ecliptic, have experienced any changes 
from secular inequalities ; for, although the mean action of 
the sun on the moon depends upon the inclination of the 
lunar orbit to the ecliptic, and the position of the ecliptic is 
subject to a secular inequality, yet analysis shows, that it 



364 ELEMENTS OF ASTRONOMY. 

does not occasion a secular variation in the inclination of 
the lunar orbit, because the action of the sun constantly 
brings the moon's orbit to the same inclination to the 
ecliptic. 

The mean motion, the nodes, and the perigee, however, 
are subject to very remarkable variations. 

§ 519. From the eclipse observed by the Chaldeans 
at Babylon, on the 19th of March, 721 years before the 
Christian eva, the place of the moon is known from that of 
the sun at the moment of opposition, whence her mean lon- 
gitude may be fouud. But the comparison of this mean 
longitude with another mean longitude, computed back for 
the instant of the eclipse from modern observations, shows 
that the moon performs her revolution round the earth 
more rapidity and in a shorter time now than she did for- 
merly, and that the acceleration in her mean motion has 
been increasing from age to age as the square of the time. 
All ancient and intermediate eclipses confirm this result. 
As the mean motions of the planets have no secular ine- 
qualities, this seemed to be an unaccountable anomaly. It 
was at one time attributed to the resistance of an ethereal 
medium pervading space, and at another to the successive 
transmission of the gravitating force. But as La Place 
proved that neither of these causes, even if they exist, have 
any influence on the motions of the lunar perigee or nodes, 
they could not affect the mean motion ; a variation in the 
mean motion from such causes being inseparably connected 
with variations in the motions of the perigee and nodes. 
He perceived that the secular variation in the elements of 
Jupiter's orbit, from the action of the planets, occasions 
corresponding changes in the motions of the satellites, 
which led him to suspect that the acceleration in the mean 
motion of the moon might be connected with the secular 
variation in the eccentricity of the terrestrial orbit. Anal- 
ysis has shewn that he assigned the true cause of the ac- 
celeration. 

§ 520. It is proved that the greater the eccentricity 
of the terrestrial orbit, the greater is the disturbing action 
of the sun on the moon. Now as the eccentricity has been 
decreasing for ages, the effect of the sun in disturbing the 



ELEMENTS OP ASTKONOMY. 365 

moon has been diminishing during that time. Consequent- 
ly the attraction of the earth has had a more and more 
powerful effect on the moon, and has been continually di- 
minishing the size of the lunar orbit. So that the moon's 
velocity has been gradually augmenting for many centuries 
to balance the increase of the earth's attraction. This 
secular increase in the moon's velocity is called the accele- 
ration, a name peculiarly appropriate at present, and which 
will continue to be so for a vast number of ages ; because, 
as long as the earth's eccentricity diminishes, the moon's 
mean motion will be accelerated ; but when the eccentricity 
has passed its minimum, and begins to increase, the mean 
motion will be retarded from age to age. The secular ac- 
celeration is now about 11". 9, but its effect on the moon's 
place increases as the square of the time. It is remarka- 
ble that the action of the planets, thus reflected by the sun 
to the moon, is much more sensible than their direct action 
either on the earth or moon. The secular diminution in 
the eccentricity, which has not altered the equation of the 
centre of the sun by eight minutes since the earliest re- 
corded eclipses, has produced a variation of about 1° 48' 
in the moon's longitude, and of 7° 12' in her mean 
anomaly. 

§ 521. The action of the sun occasions a rapid but 
variable motion in the nodes and perigee of the lunar orbit. 
Though the nodes recede during the greater part of the 
moon's revolution, and advance during the smaller, they 
perform their sidereal revolution in 6,793d. 9h. 23' 9". 3 ; 
and the perigee accomplishes a revolution in 3,232d. 13h. 
48' 29". 6, or a little more than nine years, notwithstand- 
ing its motion is sometimes retrograde and sometimes di- 
rect : but such is the difference between the disturbing 
action of the sun and that of all the planets put together, 
that it requires no less than 109,830 years for the greater 
axis of the terrestrial orbit to do the same, moving at the 
rate of 11". 8 annually. The form of the earth has no 
sensible effect either on the lunar nodes or apsides. It is 
evident that the same secular variation which changes the 
sun's distance from the earth, and occasions the accelera- 
tion of the moon's mean motion, must affect the nodes and 
31* 



366 ELEMENTS OF ASTRONOMY. 

perigee. It consequently appears, from theory as well as 
observation, that both these elements are subject to a secu- 
lar inequality, arising from the variation in the eccentricity 
of the earth's orbit, which connects them with the accelera- 
tion, so that both are retarded when the mean motion is 
anticipated. 

§ 522. The moon is so near that the excess of matter 
at the earth's equator occasions periodic variations in her 
longitude, and also that remarkable inequality in her lati- 
tude, already mentioned as a nutation in the lunar orbit, 
which diminishes its inclination to the ecliptic when the 
moon's ascending node coincides with the equinox of spring, 
and augments it when that node coincides with the equinox 
of autumn. As the cause must be proportional to the 
effect, a comparison of these inequalities, computed from 
theory, with the same given by observation, shows that the 
compression of the terrestrial spheroid, or the ratio of the 
difference between the polar and equatorial diameters, to 
the diameter of the equator, is ^-g-. It is proved ana- 
lytically, that if a fluid mass of homogeneous matter, whose 
particles attract each other inversely as the squares of the 
distance, were to revolve about an axis as the earth does, 
it would assume the form of a spheroid whose compression 
is 23 0* Since that is not the case, the earth cannot be 
homogeneous, but must decrease in density from its centre 
to its circumference. Thus the moon's eclipses show the 
earth to be round ; and her inequalities determine not only 
the form, but also the internal structure of our planet ; re- 
sults of analysis which could not have been anticipated. 
Similar inequalities in the motions of Jupiter's satellites 
prove that his mass is not homogeneous, and that his com- 
pression is x^vg-. 



ELEMENTS OF ASTKONOMY. 367 



CHAPTER XXIII. 

ECLIPSES. 

Conditions necessary for an Eclipse. Lunar Eclipses. Dimensions of the 
Earth's Shadow. Limits of a Lunar Eclipse. Solar Eclipse. Effect of 
the Moon's Parallax. Limits of a Solar Eclipse. Number of Eclipses in 
a year. Eclipse of 1706. Eclipse of 1S42. 

§ 523. Since the sun and moon are equally concerned 
in eclipses, their details have been reserved till we were 
fully acquainted with the motions of both these bodies. 
To make an eclipse three bodies are necessary, — a light- 
giving, a light-receiving, and a light-intercepting body. 
These three bodies must be wholly or partially in one 
straight line. If the moon is new and intercepts the light, 
we have a solar eclipse ; if the moon is full and the earth 
cuts off the light, we have a lunar eclipse. In the last 
case the moon is actually deprived of light by passing into 
the earth's shadow ; in the first case only a small portion 
of the earth is really deprived of light or eclipsed, but to 
that portion the w T hole or part of the sun appears eclipsed, 
though in fact it is as bright as ever. When the moon is 
eclipsed, the sun appears eclipsed to her, totally so to all 
those parts on which the earth's shadow falls, so long as 
they are in the shadow. When the sun is eclipsed to us, 
the moon's inhabitants, if she has any, see her shadow like 
a dark spot travelling over the earth, about twice as fast 
as its equatorial parts move, and in the same direction. 

§ 524. In order that either kind of an eclipse should 
happen, one body must pass into the shadow of another or 
into the line of direction of its shadow. Every planet, and 
every satellite, casts a shadow toward that point of the 
heavens which is opposite the sun. We might therefore 
expect eclipses to occur continually. But the primaries 
are too distant from one another ever to enter one another's 
shadow ; the only eclipses which can take place are be- 
tween the primaries and secondaries which are so much 



368 ELEMENTS OF ASTRONOMY. 

nearer. The occurrence of eclipses is made more frequent 
both by the greater length and the greater circumference 
of a shadow. A shadow may be cast by a large body, but 
if the one which gives light be vastly larger the shadow 
will be a short cone. If the light-giving and light-inter- 
cepting bodies were of the same size, the shadow would be 
a cylinder extending infinitely ; in this case the body would 
be equally eclipsed at whatever distance from the interven- 
ing body it passed. If the shadow were a cone, the body 
would be eclipsed longer if it passed near the base of the 
cone. Since the sun is so much larger than the planets, 
their shadows are all conical and not very long. The 
shadow of the earth where the moon passes through it is 
large enough to cover the moon's disc if its diameter were 
three times what it now is ; the moon's shadow often ends 
before it can reach the earth, and never, unless it falls 
very obliquely, covers more than 170 linear miles of its 
surface. 

If the moon's orbit coincided with the plane of the eclip- 
tic there would be a lunar eclipse every full moon, and a 
solar eclipse every new moon ; the moon would be eclipsed 
to all one hemisphere of the earth for an hour and a half; 
the sun would to certain small portions of the earth appear 
darkened for a space not exceeding eight minutes. Since 
the moon's orbit is inclined, eclipses only take place when 
the moon is new or full near her nodes. 

§ 525. We shall begin with lunar eclipses, because 
they are more simple. In considering them we need not 
refer to any particular point on the surface of the earth, 
or embarrass ourselves with observations made in different 
places, for her disappearance is absolute and universal ; 
we have only to ascertain when the light transmitted from 
the sun actually ceases to reach the moon. 

If the moon happens to be full in her node, the centre 
of her disc will pass through the centre of the earth's 
shadow ; the eclipse will be total and of the longest dura- 
tion. If the moon is full near her nodes, she may enter 
one or the other edge of the earth's shadow, and there 
may be a partial eclipse for a shorter time. If she is full 
90° from her node, her centre will be upwards of 5° from 



ELEMENTS OF ASTRONOMY. 369 

the centre of her shadow, and there -will be no eclipse. 
There must be a certain distance from her node, beyond 
which there can be no eclipse. This is called the lunar 
eclipse limit. It may be. found by calculating the size of 
the earth's shadow at the mean distance of the moon from 
the earth, and adding to its radius the radius of the moon's 
disc. If the right line joining the ecliptic and the moon's 
orbit at a given distance from the node be less than these, 
there may be an eclipse ; if more, no eclipse is possible. 
Where the distance from the moon's orbit to the ecliptic 
equals these two quantities added, is the lunar eclipse 
limit. 

§ 526. The distance of the earth from the sun and the 
sun's and the earth's sizes being known, the length of the 
earth's cone may be found by similar triangles. It extends 
220 of the earth's radii, while the moon is only 60 of such 
radii distant. The breadth of the shadow at the mean 
distance of the moon may be found by the following pro- 
portion. As 220 radii : 220—60 : : 1 : T 8 T of the earth's 
diameter, or of 7,912 miles. As the diameter of the moon 
is 2,160 miles, and as the angles are small, we may find 
the angle this shadow will subtend by the proportion, 
2,160 : 5,754 : : the angular diameter of the moon : to 
the angular diameter of the earth's shadow at the distance 
of the moon. 

The farther the sun is from the earth the more slowly 
do the boundaries of the shadow approach each other, and 
the larger therefore is the shadow at the moon's distance. 
The nearer the moon is to the earth, the larger, other 
things continuing the same, is the part of the shadow 
through which she passes. On both accounts the duration 
of an eclipse is greatest when the moon is at the least, and 
the sun at the greatest distance. These causes also make 
some difference in the limits within which an eclipse can 
take place. 

' § 527. Taking the extreme values, the greatest appa- 
rent radius of the shadow is 45' 12".15, and the corres- 
ponding apparent radius of the moon is 16' 5".45 ; and the 
greatest distance from the centre of the earth's shadow, 
at which the moon can possibly come in contact with it, is 



370 ELEMENTS OF ASTRONOMY. 

the sum of these quantities, or 62 7 37". 65. In the same 
manner the least apparent radius of the shadow is 36' 
42".15, and the correspondent apparent radius of the moon 
is 14' 41", and the least distance at which the moon 
can just be in contact with the shadow and no more, is the 
sum of these quantities, or 51' 23". 15. If therefore the 
moon never comes so near as 62' 37". 65 to the centre of 
the earth's shadow, there can be no eclipse ; if she comes 
to that distance or within it, there may ; if she conies with- 
in the distance of 51' 23". 15, there must be an eclipse. 

Let N M (Fig. 14, Plate II.) represent a portion of the 
moon's orbit, N E a portion of the ecliptic, N of course be- 
ing the node. Let E M, a secondary to Mn, be 62' 37". 
If we can ascertain what must be the value of EN to cor- 
respond with this value of E M, we shall ascertain how dis- 
tant the node ma} T be from the centre of the earth's shadow 
to admit of there being an eclipse of the moon. The angle 
EMnisa right angle, E N M is 5° 17', and the side E M 
is known by supposition. The remaining sides and angles 
may therefore be computed. E N is equal to 11° 25' 40" 
nearly. If E M is taken=51' 23", E N will equal 9° 20' 
29" nearly. An eclipse may or must take place within 
these limits on each side of the node. 

§ 528. We have hitherto spoken of the shadow as 
conical ; and it is true that the portion of space within 
which the earth will entirely conceal the sun is so. This 
conical shadow is called the umbra. But there will be 
another portion within which a part of the sun will be con- 
cealed. Beyond the umbra are her diverging spaces, 
where if a spectator be situated he will see only a portion 
of the sun's surface, the rest being obscured by the earth. 
' B C and A D, two common tangents to the sun and the 
earth, drawn crossing the line which joins their centres, 
give the limits of this faint shadow or penumbra on both 
sides. The penumbra lies on all sides of the umbra. 

In a lunar eclipse, the moon enters the penumbra first, 
and gradually gets involved in the umbra. It is difficult 
to ascertain the moment of passing from one to the other, 
and for this reason eclipses of the moon cannot give ter- 
restrial longitude exactly. When the centre of the moon 



ELEMENTS OF ASTRONOMY. 371 

passes through the centre of the shadow, the eclipse is called 
central or total. When the moon passes through the upper 
or lower portion of the shadow, the eclipse is called partial. 
In order to mark the extent of the eclipse, the diameter 
of the moon (or sun) is supposed to be divided into twelve 
equal parts, called digits, and the depth of the immersion 
is estimated in digits. When the moon enters the penum- 
bra only, it is not said to be eclipsed. 

§ 529. The consideration of a solar eclipse is more 
embarrassing. One calculation for the whole earth will 
not answer here ; the position of the spectator on the 
earth's surface, and even the rotation of the earth must be 
allowed for ; and the duration and extent of the eclipse 
must be computed for particular places. The sun is seen 
from the earth in nearly its true place, but the moon's 
parallax is considerable, and must not be neglected in find- 
ing the limits within which a solar eclipse may take place. 
By parallax the moon's apparent edge may be thrown in 
any direction according to the spectator's station, by any 
amount not exceeding the horizontal parallax. Now this 
comes to the same, so far as the possibility of an eclipse is 
concerned, as if the apparent diameter of the moon, seen 
from the earth's centre, were dilated by twice its horizon- 
tal parallax ; for if when so dilated it can touch or over 
lap the sun, there must be an eclipse at some part or other 
of the earth's surface. This sum is at its maximum about 
1° 34 ; 21". Trom this the lunar ecliptic limit is found to 
be 17° 21/ 21", when the sun is farthest and the moon 
nearest ; 15° 14' 21", when the sun is at its least and the 
moon at its greatest distance, and therefore appears smallest. 
If then at the moment of the new moon the moon's node is 
farther from the sun in longitude than this limit, there can 
be no eclipse ; if within 17°, there may ; and if within 15°, 
there must be one, to some part of the earth. To ascer- 
tain for any place whether there will be one or not, and 
also its extent, the place of the node and the semi-diameters 
must be exactly ascertained, and the local parallax, and 
also the increase in the moon's apparent diameter, owing 
to the spectator's being nearer her than the centre of the 
earth is, must be found. 



372 ELEMENTS OF ASTRONOMY. 

§ 530. When the moon is in perigee while the sun is 
in apogee, her distance from the centre of the earth is not 
quite sixty radii of the earth, her shadow reaches to the 
earth, and to some portion of the earth there will be a 
total eclipse. When the moon is in apogee while the sun 
is in perigee, her distance is nearly sixty-four radii, the 
earth is beyond the termination of the shadow, and there 
can be no total eclipse, though there may be an annular 
one, or one in which the moon covers the central part of 
the sun's disc and leaves a ring-shaped surface visible. 
Intermediate positions of the sun and moon require corres- 
ponding calculations ; in general when the apparent diame- 
ter of the moon exceeds that of the sun, the eclipse is total, 
when the sun's is the largest, it is annular. 

The portion of the earth's surface at which the eclipse is 
total at the same moment cannot exceed a circle with a 
radius of eighty-eight miles, if the centre of the earth is in 
a line with the axis of the shadow. If the centre of the 
earth be not in this line, the radius of the shadow will be 
less, but the shadow will fall more obliquely. 

The whole region of the earth, however, to which the 
eclipse may be total is greater than has been stated. The 
motion of the moon carries her shadow along a zone or 
belt of the earth's surface, and the eclipse is total, though 
at different times to the inhabitants of different parts within 
this belt. Thus the time at which a total eclipse takes 
place is different for different places. Its duration will 
also be different as the centre of the shadow, or only a 
more remote part of it, passes over the spot. It can never 
continue total at any particular place for more than V 38", 
nor be annular for more than 12 ; 24". An annular eclipse 
must be longer than a total one, because the sun's diame- 
ter being the largest the moon occupies more time in 
traversing it. In partial eclipses the observer is only with- 
in the penumbra of the moon, and more or less of the sun 
is hidden, as the observer is less or more remote from the 
centre of the sun. The penumbra covers a circle or ellipse 
of about 2,000 miles radius. 

§ 531, The duration of eclipses is modified by the 
motions of the moon and of the earth. Owing to the moon 



ELEMENTS OF ASTRONOMY. 373 

and earth's motion being round the sun in the same direc- 
tion, lunar and solar eclipses are slightly lengthened, the 
moon keeps in the earth's shadow longer than if the earth 
stood still. The moon's revolution round the earth shortens 
by a minute quantity solar eclipses. The varying rapidity 
with which the earth and moon move in different parts of 
their orbits, introduces another cause of irregularity into 
eclipses. The greatest total eclipse is shortened because 
the moon is then in her perigee ; annular eclipses are 
lengthened because the moon is then in apogee. The mo- 
tion of the earth in rotation has no effect on lunar eclipses. 
Since the moon passes the same way, but twice as fast, a 
solar eclipse is lengthened a little to most places. But if 
the earth is so inclined that the eclipse extends over one 
of its poles, an observer in that region would be carried 
through the shadow more quickly than if he were carried 
by the earth's rotation. 

§ 532. As the solar eclipse limits exceed the lunar, 
there must be more eclipses of the sun than of the moon. 
But every eclipse of the moon is visible wherever the moon 
is above the horizon at the time when it takes place, that 
is to half the earth ; and as she is above the horizon of 
each particular place as long during the year as she is be- 
low it, half of her eclipses are visible to each observer 
wherever situated. Not only do half of the eclipses of the 
sun take place while he is below the horizon of a certain 
place, but, as we have seen, it may not be visible at many 
places while he is above the horizon. Thus, though the 
whole number of solar eclipses exceeds that of the lunar, 
the number visible at any particular place falls short of it. 

Since the solar eclipse limits are from 30° to 34°, there 
must be at least one new moon and perhaps two while the 
sun is so near the node as to be eclipsed. This may hap- 
pen at both nodes in one year ; and as the nodes retreat 
19° in the course of a year, the sun may again come round 
so near the node as to be eclipsed. Hence there may be 
five solar eclipses in one year. Since the lunar eclipse 
limits do not exceed 22°, and the sun is less than a month 
in moving through that space, there may not be any full 
moon near the node, and consequently no lunar eclipse. 
32 



374 ELEMENTS OF ASTRONOMY. 

But if there are two solar eclipses at one node, there must 
be one lunar between them. And also if there be one 
lunar eclipse near a node there must be at least one solar 
eclipse also. The greatest number of eclipses which can 
take place in a year is seven ; five solar, and two lunar. 
The least number is two, both of which will be solar. 
When there are four solar, there may, owing to the motion 
of the nodes, be three lunar ; but when there are five solar, 
there can be but two lunar. 

§ 533. After a certain period eclipses return very 
nearly in the same order and of the same magnitude. 
223 of the moon's synodical revolutions occupy 6585.32 
days, and nineteen complete synodical revolutions of the 
node occupy 6585.78. After this period eclipses return 
very nearly in the same order as before, though not so ac- 
curately as to dispense with the trouble of calculating them. 
The period of eighteen years and ten days was early dis- 
covered by the Chaldeans and used by them in foretelling 
eclipses. Another period, called the Metonic cycle, con- 
sists of 235 synodical revolutions, or 19 tropical years. 
New and full moons fall on the same day in every Metonic 
cycle ; it has therefore been much used in regulating games 
and feasts and fasts. By means of these cycles, and our 
knowledge of the laws of eclipses, the date of some histori- 
cal events has been fixed with great precision. The date 
of the battle of Arbela has been determined from its being 
fought eleven days after an eclipse whose period has been 
computed. And a battle between the Medes and the 
Lydians, which was broken off in consequence of an eclipse, 
and of which even the year was not known, has been simi- 
larly investigated, for there was only one eclipse about 
that time which could be total in the part of Asia where 
it was fought. 

§ 534. Since so many circumstances are necessary to 
produce total or annular eclipses, their occurrence at any 
one place is extremely rare. The eclipse of 1706 was total 
for a long time over a great extent of country in Europe, 
from Seville, crossing Spain diagonally, the southern parts 
of France, part of Switzerland and Germany, Poland, and 
the countries to the northeast, even to the Frozen Ocean. 



ELEMENTS OE ASTRONOMY. 375 

At Montpellier total darkness lasted nearly five minutes. 
The obscurity resembled neither real night nor twilight. 
Planets and stars were to be seen. The affrighted animals 
deserted their pastures, and sought their stables ; birds of 
night left their retreats, and those of the day sought for 
shelter. Round the obscure disc of the moon was a lumi- 
nous ring which becoming fainter, extended 4° 30 ; on all 
sides. It was not however a ring of the solar disc, for the 
apparent diameter of the moon exceeded that of the sun by 
more than two minutes. The light also was much more 
pale and gloomy than that of the sun in an annular eclipse. 
§ 535. In 1842 another total eclipse of the sun, visi- 
ble on the continent of Europe, occurred. Long journeys 
were undertaken by distinguished astronomers from the de- 
sire to view this rare phenomenon. Before leaving home 
they had agreed not to interchange one word on the sub- 
ject of the eclipse till each had written and published his 
own observations. Each one, on his return, described in 
the same manner the striking peculiarities of this eclipse, 
the luminous ring, which must have been the sun's atmo- 
sphere, surrounding the dark disc of the moon, the burst- 
ing out of three large red protuberances or tongues of 
flame from the edge of the moon, evidently connected with 
this ring or crown of light, and the rainbow changes of 
color which these protuberances took on. These tongues 
of light remained unchanged in shape during the eclipse 
after they had once appeared, proving themselves to be 
realities and not an optical illusion. This eclipse was not 
only interesting from the great beauty of the corona and 
the rose-colored lights, but it seems also to prove the non- 
existence of an atmosphere in the moon, and the existence 
of one round the sun. Arago infers that the moon has no 
atmosphere from the perfectly well-defined horns of the 
sun's crescent. If the moon had ever so thin an atmo- 
sphere, the rays from these horns passing over the dark 
portion of the moon before they met the earth, would be 
deflected, and the outline of the crescent would be injured ; 
but nothing of this kind was seen. The luminous ring did 
not belong to the moon, for it did not move with it ; on the 
contrary, the moon appeared to glide in front of it and of 



376 ELEMENTS OF ASTRONOMY. 

the colored heights. The ring was described by one ob- 
server as one third, by another as one eighth part of the 
sun's diameter. Sir John Herschel considers the tongues, 
which were by different observers variously described as 
naming or icy mountains, to be rose-colored clouds floating 
in the sun's atmosphere. 

§ 536. The effect of the eclipse upon the population 
of Perpignan, who were watching it, is described by M. 
Arago as singular and even affecting. The gravest per- 
sons were unable to restrain expressions of joy when the 
sun re-appeared ; and whilst the eclipse lasted, anxiety was 
depicted on every countenance. The effect upon animals 
was remarkable. One of the friends of M. Arago had 
placed five healthy linnets in a cage. During the sudden 
darkness of the eclipse, three of the five died. The oxen 
formed into a circle, with their horns thrust forward, as if 
preparing for the attack of an enemy. At Montpellier, 
bats and owls left their retreats, and sheep laid down as 
for the night, and the horses in the fields were in a state of 
terror. In addition to these facts, it is said, that a swarm 
of ants in full march stopped short at the moment of occul- 
tation. 

At Paria, over which town the line of central darkness 
exactly passed, at the moment when the total obscuration 
commenced, a brilliant crown of glory encircled the moon, 
like the aureola, which painters append to saints. Sud- 
denly, from the border of the black and laboring moon, 
thus singularly enshrined, burst forth at three distinct 
points, within the aureola, purple or blue flames, visible to 
every eye. At this moment, from the whole assembled 
population of the town, a simultaneous and deafening shout 
broke forth. 



Plate TL. 

Fig. 2. 

B A^ C 




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